openpiv package¶

Submodules¶

openpiv.PIV_3D_plotting module¶

functions to plot 3D-deformation fields and simple 3D-structures

openpiv.PIV_3D_plotting.explode(data)[source]¶

openpiv.PIV_3D_plotting.plot_3D_alpha(data)[source]¶

openpiv.PIV_3D_plotting.quiver_3D(u, v, w, x=None, y=None, z=None, mask_filtered=None, filter_def=0, filter_reg=(1, 1, 1), cmap='jet', quiv_args=None, vmin=None, vmax=None, arrow_scale=0.15, equal_ax=True)[source]¶

Displaying 3D deformation fields vector arrows

Parameters:

u,v,w (3d ndarray or lists) – arrays or list with deformation in x,y and z direction
x,y,z (3d ndarray or lists) – Arrays or list with deformation the coordinates of the deformations. Must match the dimensions of the u,v qnd w. If not provided x,y and z are created with np.indices(u.shape)
boolean 3d ndarray or 1d ndarray (mask_filtered,) – Array, or list with same dimensions as the deformations. Defines the area where deformations are drawn
filter_def (float) – Filter that prevents the display of deformations arrows with length < filter_def
filter_reg (tuple,list or int) – Filter that prevents the display of every i-th deformations arrows separatly alon each axis. filter_reg=(2,2,2) means that only every second arrow along x,y z axis is displayed leading to a total reduction of displayed arrows by a factor of 8. filter_reg=3 is interpreted as (3,3,3).
cmap (string) – matplotlib colorbar that defines the coloring of the arrow
quiv_args (dict) – Dictionary with kwargs passed on to the matplotlib quiver function.
vmin,vmax (float) – Upper and lower bounds for the colormap. Works like vmin and vmax in plt.imshow().

arrow_scale: float: Automatic scaling of the quiver arrows so that the longest arrow has the length axis length * arrow_scale. Arrow length can alternatively be set by passing a “lenght” argument in quiv_args.
equal_axes: bool: resize the figure axis so that they are have equal scaling.

Returns:	fig (matploltib figure object) ax (mattplotlib axes object) – the holding the main 3D quiver plot

openpiv.PIV_3D_plotting.scatter_3D(a, cmap='jet', sca_args=None, control='color', size=60)[source]¶

openpiv.PIV_3D_plotting.set_axes_equal(ax)[source]¶: Following https://stackoverflow.com/questions/13685386/matplotlib-equal-unit-length-with-equal-aspect-ratio-z-axis-is-not-equal-to Make axes of 3D plot have equal scale so that spheres appear as spheres, cubes as cubes, etc.. This is one possible solution to Matplotlib’s ax.set_aspect(‘equal’) and ax.axis(‘equal’) not working for 3D.

Parameters

ax: matplotlib.axes object

openpiv.filters module¶

The openpiv.filters module contains some filtering/smoothing routines.

openpiv.filters.gaussian(u: numpy.ndarray, v: numpy.ndarray, half_width: int = 1) → Tuple[numpy.ndarray, numpy.ndarray][source]¶

Smooths the velocity field with a Gaussian kernel.

Parameters:

u (2d np.ndarray) – the u velocity component field
v (2d np.ndarray) – the v velocity component field
half_width (int) – the half width of the kernel. Kernel has shape 2*half_width+1, default = 1

Returns:

uf (2d np.ndarray) – the smoothed u velocity component field
vf (2d np.ndarray) – the smoothed v velocity component field

openpiv.filters.gaussian_kernel(sigma: float, truncate: float = 4.0) → numpy.ndarray[source]¶: Return Gaussian that truncates at the given number of standard deviations.

openpiv.filters.replace_outliers(u: numpy.ndarray, v: numpy.ndarray, flags: numpy.ndarray, w: Optional[numpy.ndarray] = None, method: str = 'localmean', max_iter: int = 5, tol: float = 0.001, kernel_size: int = 1) → Tuple[numpy.ndarray, ...][source]¶

Replace invalid vectors in an velocity field using an iterative image: inpainting algorithm.

The algorithm is the following:

For each element in the arrays of the u and v components, replace it by a weighted average of the neighbouring elements which are not invalid themselves. The weights depends of the method type. If method=localmean weight are equal to 1/( (2*kernel_size+1)**2 -1 )
Several iterations are needed if there are adjacent invalid elements. If this is the case, inforation is “spread” from the edges of the missing regions iteratively, until the variation is below a certain threshold.

Parameters:

u (2d or 3d np.ndarray) – the u velocity component field
v (2d or 3d np.ndarray) – the v velocity component field
w (2d or 3d np.ndarray) – the w velocity component field
flags (2d array of positions with invalid vectors) –
grid_mask (2d array of positions masked by the user) –
max_iter (int) – the number of iterations
kernel_size (int) – the size of the kernel, default is 1
method (str) – the type of kernel used for repairing missing vectors

Returns:

uf (2d or 3d np.ndarray) – the smoothed u velocity component field, where invalid vectors have been replaced
vf (2d or 3d np.ndarray) – the smoothed v velocity component field, where invalid vectors have been replaced
wf (2d or 3d np.ndarray) – the smoothed w velocity component field, where invalid vectors have been replaced

openpiv.lib module¶

openpiv.lib.get_dist(kernel, kernel_size)[source]¶

openpiv.lib.replace_nans(array, max_iter, tol, kernel_size=2, method='disk')[source]¶

Replace NaN elements in an array using an iterative image inpainting: algorithm.

The algorithm is the following:

For each element in the input array, replace it by a weighted average of the neighbouring elements which are not NaN themselves. The weights depend on the method type. See Methods below.
Several iterations are needed if there are adjacent NaN elements. If this is the case, information is “spread” from the edges of the missing regions iteratively, until the variation is below a certain threshold.

Methods:

localmean - A square kernel where all elements have the same value,

weights are equal to n/( (2*kernel_size+1)**2 -1 ), where n is the number of non-NaN elements.

disk - A circular kernel where all elements have the same value,

kernel is calculated by::

if ((S-i)**2 + (S-j)**2)**0.5 <= S:: kernel[i,j] = 1.0
else:: kernel[i,j] = 0.0

where S is the kernel radius.

distance - A circular inverse distance kernel where elements are

weighted proportional to their distance away from the center of the kernel, elements farther away have less weight. Elements outside the specified radius are set to 0.0 as in ‘disk’, the remaining of the weights are calculated as:

maxDist = ((S)**2 + (S)**2)**0.5
kernel[i,j] = -1*(((S-i)**2 + (S-j)**2)**0.5 - maxDist)

where S is the kernel radius.

Parameters:	array (2d or 3d np.ndarray) – an array containing NaN elements that have to be replaced if array is a masked array (numpy.ma.MaskedArray), then the mask is reapplied after the replacement max_iter (int) – the number of iterations tol (float) – On each iteration check if the mean square difference between values of replaced elements is below a certain tolerance tol kernel_size (int) – the size of the kernel, default is 1 method (str) – the method used to replace invalid values. Valid options are localmean, disk, and distance.
Returns:	filled – a copy of the input array, where NaN elements have been replaced.
Return type:	2d or 3d np.ndarray

openpiv.phase_separation module¶

A module for separating solid phase from liquid tracers using image processing techniques.

openpiv.phase_separation.get_particles_size_array(original_image, blur_kernel_size=1, I_sat=230, opening_ksize=3)[source]¶

Returns the array of particle image areas in pixels.

Used as a quick means to set size limits in Kalitov-Longmire method.

Usage Example:

plt.hist( get_particles_size_array(image) ) plt.title(‘Particle size distribution’)

Parameters:	original_image (np.ndarray) – Original two-phase input image blur_kernel_size (int) – Stencil width for pre-processing blur. Must be an odd number. I_sat (int) – Saturation intensity for object pixels detection process. opening_ksize (int) – Stencil width for opening operation used to remove tiny regions from object pixels. Set to -1 to skip opening.
Returns:	size_array – Array of length N, containing areas of particle regions number 0 to N in pixels.
Return type:	np.array

openpiv.phase_separation.get_size_brightness_map(original_image, blur_kernel_size=1, I_sat=230, opening_ksize=3, MAX_PARTICLE_SIZE=400)[source]¶

Returns the size-brightness map.

Used as an advanced means to set size and brightness limits in Kalitov-Longmire method.

Usage Example:

plt.imshow(im, origin=’lower’) plt.xlabel(‘Brightness’) plt.ylabel(‘Size (px)’) plt.title(‘Signal density’)

Parameters:	original_image (np.ndarray) – Original two-phase input image blur_kernel_size (int) – Stencil width for pre-processing blur. Must be an odd number. I_sat (int) – Saturation intensity for object pixels detection process. opening_ksize (int) – Stencil width for opening operation used to remove tiny regions from object pixels. Set to -1 to skip opening. MAX_PARTICLE_SIZE (int) – Particle area upper limit (Y-axis max in the map) in pixels.
Returns:	density_map – Density map (D) where D[i,j] is log10(sizebrightnessnumber) at size=i & brightness=j. See Kalitov & Longmire, 2002 for more information.
Return type:	np.ndarray

openpiv.phase_separation.khalitov_longmire(original_image, big_particles_criteria, small_particles_criteria, blur_kernel_size=1, I_sat=230, opening_ksize=3)[source]¶

Extract separated images using the method proposed by Khalitov & Longmire, 2002.

For detailed information see:

Khalitov, D., Longmire, E. Simultaneous two-phase PIV by two-parameter phase discrimination. Experiments in Fluids 32, 252–268 (2002). https://doi.org/10.1007/s003480100356

Parameters:

original_image (np.ndarray) – Original two-phase input image
big_particles_criteria – ‘min_size’ : int, [‘max_size’ : int,] [‘min_brightness’ : int,] [‘max_brightness’ : int]
} –
dictionary defining big particles criteria. 'min_size' is mandatory. (A) –
small_particles_criteria – [‘min_size’ : int,] ‘max_size’ : int, [‘min_brightness’ : int,] [‘max_brightness’ : int]
} –
dictionary defining small particles criteria. 'max_size' is mandatory. (A) –
blur_kernel_size (int) – Stencil width for pre-processing blur. Must be an odd number.
I_sat (int) – Saturation intensity for object pixels detection process.
opening_ksize (int) – Stencil width for opening operation used to remove tiny regions from object pixels. Set to -1 to skip opening.

Returns:

big_particles_img (np.ndarray) – Extracted image of the phase with bigger particles (dispersed phase)
small_particles_img (np.ndarray) – Extracted image of the phase with smaller particles (carrier phase)

openpiv.phase_separation.khalitov_longmire_analyse_particle_segments(original_image, object_pixels)[source]¶: Private function

openpiv.phase_separation.khalitov_longmire_get_object_pixels(original_image, blur_kernel_size=1, I_sat=230, opening_ksize=3)[source]¶: Private function

openpiv.phase_separation.median_filter_method(original_image, kernel_size)[source]¶

Extract separated images using a median filter

Proposed by Kiger & Pan. Original paper:

Kiger, K. T., & Pan, C. (2000). PIV technique for the simultaneous measurement of dilute two-phase flows. Journal of Fluids Engineering, Transactions of the ASME, 122(4), 811–818. https://doi.org/10.1115/1.1314864

Parameters:

original_image (np.ndarray) – Original two-phase input image
kernel_size (int) – Filter stencil width (must be an odd number). Denoted by Nf in Kiger & Pan.

Returns:

big_particles_img (np.ndarray) – Extracted image of the phase with bigger particles (dispersed phase)
small_particles_img (np.ndarray) – Extracted image of the phase with smaller particles (carrier phase)

openpiv.phase_separation.opening_method(original_image, kernel_size, iterations=1, thresh_factor=1.1)[source]¶

Extract separated images based on particle size.

This method uses an erosion filter followed by a dilation (aka opening), to remove small particle traces, leaving only bigger particles in the image.

The image of small particles is also generated by using the big particles image as a mask on the original image. A threshold process is used to intensify the edges of particles in the mask.

Parameters:

original_image (np.ndarray) – Original two-phase input image
kernel_size (int) – Erosion/dilation stencil width
iterations (int) – Erosion iterations, default = 1
thresh_factor (float) –
Used to mask big particles, default = 1.1 Mask condition is defined as :

pixel value > thresh_factor * local average intensity

Returns:

big_particles_img (np.ndarray) – Extracted image of the phase with bigger particles (dispersed phase)
small_particles_img (np.ndarray) – Extracted image of the phase with smaller particles (carrier phase)

openpiv.piv module¶

openpiv.piv.piv_example()[source]¶

PIV example uses examples/test5 vortex PIV data to show the main principles

piv(im1,im2) will create a tmp.vec file with the vector filed in pix/dt (dt=1) from two images, im1,im2 provided as full path filenames (TIF is preferable)

openpiv.piv.simple_piv(im1, im2, plot=True)[source]¶

Simplest PIV run on the pair of images using default settings

piv(im1,im2) will create a tmp.vec file with the vector filed in pix/dt (dt=1) from two images, im1,im2 provided as full path filenames (TIF is preferable, whatever imageio can read)

openpiv.preprocess module¶

This module contains image processing routines that improve images prior to PIV processing.

openpiv.preprocess.contrast_stretch(img, lower_limit=2, upper_limit=98)[source]¶

Simple percentile-based contrast stretching

Parameters:	img (image) – a two dimensional array of float32 or float64, but can be uint16, uint8 or similar type lower_limit (int) – lower percentile limit upper_limit (int) – upper percentile limit
Returns:	img – a filtered two dimensional array of the input image
Return type:	image

openpiv.preprocess.dynamic_masking(image, method='edges', filter_size=7, threshold=0.005)[source]¶

Dynamically masks out the objects in the PIV images

Parameters:

image (image) – a two dimensional array of uint16, uint8 or similar type
method (string) – ‘edges’ or ‘intensity’: ‘edges’ method is used for relatively dark and sharp objects, with visible edges, on dark backgrounds, i.e. low contrast ‘intensity’ method is useful for smooth bright objects or dark objects or vice versa, i.e. images with high contrast between the object and the background
filter_size (integer) – a scalar that defines the size of the Gaussian filter
threshold (float) – a value of the threshold to segment the background from the object default value: None, replaced by sckimage.filter.threshold_otsu value

Returns:

image (array of the same datatype as the incoming image with the)
object masked out
as a completely black region(s) of zeros (integers or floats).

Example

frame_a = openpiv.tools.imread( ‘Camera1-001.tif’ ) imshow(frame_a) # original

frame_a = dynamic_masking(frame_a,method=’edges’,filter_size=7, threshold=0.005) imshow(frame_a) # masked

openpiv.preprocess.gen_lowpass_background(img_list, sigma=3, resize=None)[source]¶

Generate a background by averaging a low pass of all images in an image list. Apply by subtracting generated background image.

Parameters:	img_list (list) – list of image directories sigma (float) – sigma of the gaussian filter resize (int or float) – disabled by default, normalize array and set value to user selected max pixel intensity
Returns:	img – a mean of all low-passed images
Return type:	image

openpiv.preprocess.gen_min_background(img_list, resize=255)[source]¶

Generate a background by averaging the minimum intensity of all images in an image list. Apply by subtracting generated background image.

Parameters:	img_list (list) – list of image directories resize (int or float) – disabled by default, normalize array and set value to user selected max pixel intensity
Returns:	img – a mean of all images
Return type:	image

openpiv.preprocess.high_pass(img, sigma=5, clip=False)[source]¶

Simple high pass filter

Parameters:	img (image) – a two dimensional array of float32 or float64, but can be uint16, uint8 or similar type sigma (float) – sigma value of the gaussian filter
Returns:	img – a filtered two dimensional array of the input image
Return type:	image

openpiv.preprocess.instensity_cap(img, std_mult=2)[source]¶

Simple intensity capping.

Parameters:	img (image) – a two dimensional array of float32 or float64, but can be uint16, uint8 or similar type std_mult (int) – how strong the intensity capping is. Lower values yields a lower threshold
Returns:	img – a filtered two dimensional array of the input image
Return type:	image

openpiv.preprocess.intensity_clip(img, min_val=0, max_val=None, flag='clip')[source]¶

Simple intensity clipping

Parameters:	img (image) – a two dimensional array of float32 or float64, but can be uint16, uint8 or similar type min_val (int or float) – min allowed pixel intensity max_val (int or float) – min allowed pixel intensity flag (str) – one of two methods to set invalid pixels intensities
Returns:	img – a filtered two dimensional array of the input image
Return type:	image

openpiv.preprocess.local_variance_normalization(img, sigma_1=2, sigma_2=1, clip=True)[source]¶

Local variance normalization by two gaussian filters. This method is used by common commercial softwares

Parameters:	img (image) – a two dimensional array of float32 or float64, but can be uint16, uint8 or similar type sigma_1 (float) – sigma value of the first gaussian low pass filter sigma_2 (float) – sigma value of the second gaussian low pass filter clip (bool) – set negative pixels to zero
Returns:	img – a filtered two dimensional array of the input image
Return type:	image

openpiv.preprocess.mask_coordinates(image_mask, tolerance=1.5, min_length=10, plot=False)[source]¶

Creates set of coordinates of polygons from the image mask

Inputs: mask : binary image of a mask.

[tolerance] : float - tolerance for approximate_polygons, default = 1.5

[min_length] : int - minimum length of the polygon, filters out the small polygons like noisy regions, default = 10

Outputs:

mask_coord : list of mask coordinates in pixels

Example

# if masks of image A and B are slightly different: image_mask = np.logical_and(image_mask_a, image_mask_b) mask_coords = mask_coordinates(image_mask)

openpiv.preprocess.normalize_array(array, axis=None)[source]¶

Min/max normalization to [0,1].

Parameters:	array (np.ndarray) – array to normalize axis (int, tuple) – axis to find values for normalization
Returns:	array – normalized array
Return type:	np.ndarray

openpiv.preprocess.offset_image(img, offset_x, offset_y, pad='zero')[source]¶

Offset an image by padding.

Parameters:	img (image) – a two dimensional array of float32 or float64, but can be uint16, uint8 or similar type offset_x (int) – offset an image by integer values. Positive values shifts the image to the right and negative values shift to the left offset_y (int) – offset an image by integer values. Positive values shifts the image to the top and negative values shift to the bottom pad (str) – pad the shift with zeros or a reflection of the shift
Returns:	img – a transformed two dimensional array of the input image
Return type:	image

openpiv.preprocess.prepare_mask_from_polygon(x, y, mask_coords)[source]¶

Converts mask coordinates of the image mask to the grid of 1/0 on the x,y grid Inputs:

x,y : grid of x,y points mask_coords : array of coordinates in pixels of the image_mask

Outputs:: grid of points of the mask, of the shape of x

openpiv.preprocess.prepare_mask_on_grid(x: numpy.ndarray, y: numpy.ndarray, image_mask: numpy.ndarray) → numpy.array[source]¶

_summary_

Parameters:	x (np.ndarray) – x coordinates of vectors in pixels y (np.ndarray) – y coordinates of vectors in pixels image_mask (np.ndarray) – image of the mask, 1 or True is to be masked
Returns:	boolean array of the size of x,y with 1 where the values are masked
Return type:	np.ndarray

openpiv.preprocess.standardize_array(array, axis=None)[source]¶

Standardize an array.

Parameters:	array (np.ndarray) – array to normalize axis (int, tuple) – axis to find values for standardization
Returns:	array – normalized array
Return type:	np.ndarray

openpiv.preprocess.stretch_image(img, x_axis=0, y_axis=0)[source]¶

Stretch an image by interplation.

Parameters:	img (image) – a two dimensional array of float32 or float64, but can be uint16, uint8 or similar type x_axis (float) – stretch the x-axis of an image where 0 == no stretching y_axis (float) – stretch the y-axis of an image where 0 == no stretching
Returns:	img – a transformed two dimensional array of the input image
Return type:	image

openpiv.preprocess.threshold_binarize(img, threshold, max_val=255)[source]¶

Simple binarizing threshold

Parameters:	img (image) – a two dimensional array of float32 or float64, but can be uint16, uint8 or similar type threshold (int or float) – boundary where pixels set lower than the threshold are set to zero and values higher than the threshold are set to the maximum user selected value max_val (int or float) – maximum pixel value of the image
Returns:	img – a filtered two dimensional array of the input image
Return type:	image

openpiv.process module¶

openpiv.pyprocess module¶

This module contains a pure python implementation of the basic cross-correlation algorithm for PIV image processing.

openpiv.pyprocess.correlate_windows(window_a, window_b, correlation_method='fft', convolve2d=<function convolve2d>, rfft2=<function rfft2>, irfft2=<function irfft2>)[source]¶

Compute correlation function between two interrogation windows. The correlation function can be computed by using the correlation theorem to speed up the computation. :param window_a: a two dimensions array for the first interrogation window :type window_a: 2d np.ndarray :param window_b: a two dimensions array for the second interrogation window :type window_b: 2d np.ndarray :param correlation_method: ‘circular’ - FFT based without zero-padding

‘linear’ - FFT based with zero-padding ‘direct’ - linear convolution based Default is ‘fft’, which is much faster.

Parameters:

convolve2d (function) – function used for 2d convolutions
rfft2 (function) – function used for rfft2
irfft2 (function) – function used for irfft2

Returns:

corr (2d np.ndarray) – a two dimensions array for the correlation function.
Note that due to the wish to use 2^N windows for faster FFT
we use a slightly different convention for the size of the
correlation map. The theory says it is M+N-1, and the
’direct’ method gets this size out
the FFT-based method returns M+N size out, where M is the window_size
and N is the search_area_size
It leads to inconsistency of the output

openpiv.pyprocess.correlation_to_displacement(corr, n_rows, n_cols, subpixel_method='gaussian')[source]¶

Correlation maps are converted to displacement for each interrogation window using the convention that the size of the correlation map is 2N -1 where N is the size of the largest interrogation window (in frame B) that is called search_area_size Inputs:

corr : 3D nd.array

contains output of the fft_correlate_images

n_rows, n_cols : number of interrogation windows, output of the

get_field_shape

openpiv.pyprocess.extended_search_area_piv(frame_a: numpy.ndarray, frame_b: numpy.ndarray, window_size: Union[int, Tuple[int, int]], overlap: Union[int, Tuple[int, int]] = (0, 0), dt: float = 1.0, search_area_size: Union[int, Tuple[int, int], None] = None, correlation_method: str = 'circular', subpixel_method: str = 'gaussian', sig2noise_method: Optional[str] = 'peak2mean', width: int = 2, normalized_correlation: bool = False, use_vectorized: bool = False)[source]¶

Standard PIV cross-correlation algorithm, with an option for extended area search that increased dynamic range. The search region in the second frame is larger than the interrogation window size in the first frame. For Cython implementation see openpiv.process.extended_search_area_piv

This is a pure python implementation of the standard PIV cross-correlation algorithm. It is a zero order displacement predictor, and no iterative process is performed.

Parameters:

frame_a (2d np.ndarray) – an two dimensions array of integers containing grey levels of the first frame.
frame_b (2d np.ndarray) – an two dimensions array of integers containing grey levels of the second frame.
window_size (int) – the size of the (square) interrogation window, [default: 32 pix].
overlap (int) – the number of pixels by which two adjacent windows overlap [default: 16 pix].
dt (float) – the time delay separating the two frames [default: 1.0].
correlation_method (string) – one of the two methods implemented: ‘circular’ or ‘linear’, default: ‘circular’, it’s faster, without zero-padding ‘linear’ requires also normalized_correlation = True (see below)
subpixel_method (string) – one of the following methods to estimate subpixel location of the peak: ‘centroid’ [replaces default if correlation map is negative], ‘gaussian’ [default if correlation map is positive], ‘parabolic’.
sig2noise_method (string) – defines the method of signal-to-noise-ratio measure, (‘peak2peak’ or ‘peak2mean’. If None, no measure is performed.)
width (int) – the half size of the region around the first correlation peak to ignore for finding the second peak. [default: 2]. Only used if sig2noise_method==peak2peak.
search_area_size (int) – the size of the interrogation window in the second frame, default is the same interrogation window size and it is a fallback to the simplest FFT based PIV
normalized_correlation (bool) – if True, then the image intensity will be modified by removing the mean, dividing by the standard deviation and the correlation map will be normalized. It’s slower but could be more robust

Returns:

u (2d np.ndarray) – a two dimensional array containing the u velocity component, in pixels/seconds.
v (2d np.ndarray) – a two dimensional array containing the v velocity component, in pixels/seconds.
sig2noise (2d np.ndarray, ( optional: only if sig2noise_method != None )) – a two dimensional array the signal to noise ratio for each window pair.

The implementation of the one-step direct correlation with different size of the interrogation window and the search area. The increased size of the search areas cope with the problem of loss of pairs due to in-plane motion, allowing for a smaller interrogation window size, without increasing the number of outlier vectors.

See:

Particle-Imaging Techniques for Experimental Fluid Mechanics

Annual Review of Fluid Mechanics Vol. 23: 261-304 (Volume publication date January 1991) DOI: 10.1146/annurev.fl.23.010191.001401

originally implemented in process.pyx in Cython and converted to a NumPy vectorized solution in pyprocess.py

openpiv.pyprocess.fft_correlate_images(image_a: numpy.ndarray, image_b: numpy.ndarray, correlation_method: str = 'circular', normalized_correlation: bool = True, conj: Callable = <ufunc 'conjugate'>, rfft2=<function rfft2>, irfft2=<function irfft2>, fftshift=<function fftshift>) → numpy.ndarray[source]¶

FFT based cross correlation of two images with multiple views of np.stride_tricks() The 2D FFT should be applied to the last two axes (-2,-1) and the zero axis is the number of the interrogation window This should also work out of the box for rectangular windows. :param image_a: and two last dimensions are interrogation windows of the first image :type image_a: 3d np.ndarray, first dimension is the number of windows, :param image_b: :type image_b: similar :param correlation_method: one of the three methods implemented: ‘circular’ or ‘linear’

[default: ‘circular].

Parameters:	normalized_correlation (string) – decides wetehr normalized correlation is done or not: True or False [default: True]. conj (function) – function used for complex conjugate rfft2 (function) – function used for rfft2 irfft2 (function) – function used for irfft2 fftshift (function) – function used for fftshift

openpiv.pyprocess.fft_correlate_windows(window_a, window_b, rfft2=<function rfft2>, irfft2=<function irfft2>)[source]¶

FFT based cross correlation it is a so-called linear convolution based, since we increase the size of the FFT to reduce the edge effects. This should also work out of the box for rectangular windows.

Parameters:

window_a (2d np.ndarray) – a two dimensions array for the first interrogation window
window_b (2d np.ndarray) – a two dimensions array for the second interrogation window
rfft2 (function) – function used for rfft2
irfft2 (function) – function used for irfft2
from Stackoverflow (#) –
scipy import linalg (from) –
numpy as np (import) –
works for rectangular windows as well (#) –
= [[1 , 0 , 0 , 0] , [0 , -1 , 0 , 0] , [0 , 0 , 3 , 0] , (x) – [0 , 0 , 0 , 1], [0 , 0 , 0 , 1]]
= np.array(x,dtype=np.float) (x) –
= [[4 , 5] , [3 , 4]] (y) –
= np.array(y) (y) –
("conv (print) –
= np.array(x.shape) (s1) –
= np.array(y.shape) (s2) –
= s1 + s2 - 1 (size) –
= 2 ** np.ceil(np.log2(size))astype(int) (fsize) –
= tuple([slice(0, int(sz)) for sz in size]) (fslice) –
= np.fft.fft2(x , fsize) (new_x) –
= np.fft.fft2(y , fsize) (new_y) –
= np.fft.ifft2(new_x*new_y)[fslice]copy() (result) –
for my method (print("fft) –

openpiv.pyprocess.find_all_first_peaks(corr)[source]¶

Find row and column indices of the first correlation peak.

Parameters:	corr (np.ndarray) – the correlation map fof the strided images (N,K,M) where N is the number of windows, KxM is the interrogation window size
Returns:	index_list (integers, index of the peak position in (N,i,j)) peaks_max (amplitude of the peak)

openpiv.pyprocess.find_all_second_peaks(corr, width=2)[source]¶

Find row and column indices of the first correlation peak.

Parameters:

corr (np.ndarray) – the correlation map fof the strided images (N,K,M) where N is the number of windows, KxM is the interrogation window size
width (int) – the half size of the region around the first correlation peak to ignore for finding the second peak

Returns:

index_list (integers, index of the peak position in (N,i,j))
peaks_max (amplitude of the peak)

openpiv.pyprocess.find_first_peak(corr)[source]¶

Find row and column indices of the first correlation peak.

Parameters:	corr (np.ndarray) – the correlation map fof the strided images (N,K,M) where N is the number of windows, KxM is the interrogation window size
Returns:	(i,j) (integers, index of the peak position) peak (amplitude of the peak)

openpiv.pyprocess.find_second_peak(corr, i=None, j=None, width=2)[source]¶

Find the value of the second largest peak.

The second largest peak is the height of the peak in the region outside a 3x3 submatrxi around the first correlation peak.

Parameters:

corr (np.ndarray) – the correlation map.
i,j (ints) – row and column location of the first peak.
width (int) – the half size of the region around the first correlation peak to ignore for finding the second peak.

Returns:

i (int) – the row index of the second correlation peak.
j (int) – the column index of the second correlation peak.
corr_max2 (int) – the value of the second correlation peak.

openpiv.pyprocess.find_subpixel_peak_position(corr, subpixel_method='gaussian')[source]¶

Find subpixel approximation of the correlation peak.

This function returns a subpixels approximation of the correlation peak by using one of the several methods available. If requested, the function also returns the signal to noise ratio level evaluated from the correlation map.

Parameters:	corr (np.ndarray) – the correlation map. subpixel_method (string) – one of the following methods to estimate subpixel location of the peak: ‘centroid’ [replaces default if correlation map is negative], ‘gaussian’ [default if correlation map is positive], ‘parabolic’.
Returns:	subp_peak_position – the fractional row and column indices for the sub-pixel approximation of the correlation peak. If the first peak is on the border of the correlation map or any other problem, the returned result is a tuple of NaNs.
Return type:	two elements tuple

openpiv.pyprocess.get_coordinates(image_size: Tuple[int, int], search_area_size: int, overlap: int, center_on_field: bool = True) → Tuple[numpy.ndarray, numpy.ndarray][source]¶

Compute the x, y coordinates of the centers of the interrogation windows. for the SQUARE windows only, see also get_rect_coordinates

the origin (0,0) is like in the image, top left corner positive x is an increasing column index from left to right positive y is increasing row index, from top to bottom

Parameters:

image_size (two elements tuple) – a two dimensional tuple for the pixel size of the image first element is number of rows, second element is the number of columns.
search_area_size (int) – the size of the search area windows, sometimes it’s equal to the interrogation window size in both frames A and B
overlap (int = 0 (default is no overlap)) – the number of pixel by which two adjacent interrogation windows overlap.

Returns:

x (2d np.ndarray) – a two dimensional array containing the x coordinates of the interrogation window centers, in pixels.
y (2d np.ndarray) – a two dimensional array containing the y coordinates of the interrogation window centers, in pixels.

Coordinate system 0,0 is at the top left corner, positive x to the right, positive y from top downwards, i.e. image coordinate system

openpiv.pyprocess.get_field_shape(image_size: Tuple[int, int], search_area_size: Tuple[int, int], overlap: Tuple[int, int]) → Tuple[int, int][source]¶

Compute the shape of the resulting flow field.

Given the image size, the interrogation window size and the overlap size, it is possible to calculate the number of rows and columns of the resulting flow field.

Parameters:	image_size (two elements tuple) – a two dimensional tuple for the pixel size of the image first element is number of rows, second element is the number of columns, easy to obtain using .shape search_area_size (tuple) – the size of the interrogation windows (if equal in frames A,B) or the search area (in frame B), the largest of the two overlap (tuple) – the number of pixel by which two adjacent interrogation windows overlap.
Returns:	field_shape – the shape of the resulting flow field
Return type:	2-element tuple

openpiv.pyprocess.get_rect_coordinates(image_size: Tuple[int, int], window_size: Union[int, Tuple[int, int]], overlap: Union[int, Tuple[int, int]], center_on_field: bool = False)[source]¶: Rectangular grid version of get_coordinates.

openpiv.pyprocess.moving_window_array(array, window_size, overlap)[source]¶

This is a nice numpy trick. The concept of numpy strides should be clear to understand this code.

Basically, we have a 2d array and we want to perform cross-correlation over the interrogation windows. An approach could be to loop over the array but loops are expensive in python. So we create from the array a new array with three dimension, of size (n_windows, window_size, window_size), in which each slice, (along the first axis) is an interrogation window.

openpiv.pyprocess.nextpower2(i)[source]¶: Find 2^n that is equal to or greater than.

openpiv.pyprocess.normalize_intensity(window)[source]¶

Normalize interrogation window or strided image of many windows,: by removing the mean intensity value per window and clipping the negative values to zero

Parameters:	window (2d np.ndarray) – the interrogation window array
Returns:	window – the interrogation window array, with mean value equal to zero and intensity normalized to -1 +1 and clipped if some pixels are extra low/high
Return type:	2d np.ndarray

openpiv.pyprocess.sig2noise_ratio(correlation: numpy.ndarray, sig2noise_method: str = 'peak2peak', width: int = 2) → numpy.ndarray[source]¶

Computes the signal to noise ratio from the correlation map.

The signal to noise ratio is computed from the correlation map with one of two available method. It is a measure of the quality of the matching between to interrogation windows.

Parameters:	corr (3d np.ndarray) – the correlation maps of the image pair, concatenated along 0th axis sig2noise_method (string) – the method for evaluating the signal to noise ratio value from the correlation map. Can be peak2peak, peak2mean or None if no evaluation should be made. width (int, optional) – the half size of the region around the first correlation peak to ignore for finding the second peak. [default: 2]. Only used if `sig2noise_method==peak2peak`.
Returns:	sig2noise – the signal to noise ratios from the correlation maps.
Return type:	np.array

openpiv.pyprocess.sliding_window_array(image: numpy.ndarray, window_size: Tuple[int, int] = (64, 64), overlap: Tuple[int, int] = (32, 32)) → numpy.ndarray[source]¶: This version does not use numpy as_strided and is much more memory efficient. Basically, we have a 2d array and we want to perform cross-correlation over the interrogation windows. An approach could be to loop over the array but loops are expensive in python. So we create from the array a new array with three dimension, of size (n_windows, window_size, window_size), in which each slice, (along the first axis) is an interrogation window.

openpiv.pyprocess.vectorized_correlation_to_displacements(corr: numpy.ndarray, n_rows: Optional[int] = None, n_cols: Optional[int] = None, subpixel_method: str = 'gaussian', eps: float = 1e-07)[source]¶

Correlation maps are converted to displacement for each interrogation window using the convention that the size of the correlation map is 2N -1 where N is the size of the largest interrogation window (in frame B) that is called search_area_size

Parameters:	corr (3D nd.array) – contains output of the fft_correlate_images n_cols (n_rows,) – number of interrogation windows, output of the get_field_shape mask_width (int) – distance, in pixels, from the interrogation window in which correlation peaks would be flagged as invalid
Returns:	u, v – 2d array of displacements in pixels/dt
Return type:	2D nd.array

openpiv.pyprocess.vectorized_sig2noise_ratio(correlation, sig2noise_method='peak2peak', width=2)[source]¶

Computes the signal to noise ratio from the correlation map in a mostly vectorized approach, thus much faster.

The signal to noise ratio is computed from the correlation map with one of two available method. It is a measure of the quality of the matching between to interrogation windows.

Parameters:	corr (3d np.ndarray) – the correlation maps of the image pair, concatenated along 0th axis sig2noise_method (string) – the method for evaluating the signal to noise ratio value from the correlation map. Can be peak2peak, peak2mean or None if no evaluation should be made. width (int, optional) – the half size of the region around the first correlation peak to ignore for finding the second peak. [default: 2]. Only used if sig2noise_method==peak2peak.
Returns:	sig2noise – the signal to noise ratios from the correlation maps.
Return type:	np.array

openpiv.pyprocess3D module¶

openpiv.pyprocess3D.check_input(window_size, overlap, search_area_size, frame_a, frame_b)[source]¶: check the inputs for validity

openpiv.pyprocess3D.correlate_windows(window_a, window_b, correlation_method='fft', nfftx=None, nffty=None, nfftz=None)[source]¶

Compute correlation function between two interrogation windows.

The correlation function can be computed by using the correlation theorem to speed up the computation.

Parameters:

window_a (2d np.ndarray) – a two dimensions array for the first interrogation window,
window_b (2d np.ndarray) – a two dimensions array for the second interrogation window.
correlation_method (string) – one method is currently implemented: ‘fft’.
nfftx (int) – the size of the 2D FFT in x-direction, [default: 2 x windows_a.shape[0] is recommended].
nffty (int) – the size of the 2D FFT in y-direction, [default: 2 x windows_a.shape[1] is recommended].
nfftz (int) – the size of the 2D FFT in z-direction, [default: 2 x windows_a.shape[2] is recommended].

Returns:

corr (3d np.ndarray) – a three dimensional array of the correlation function.
Note that due to the wish to use 2^N windows for faster FFT
we use a slightly different convention for the size of the
correlation map. The theory says it is M+N-1, and the
’direct’ method gets this size out
the FFT-based method returns M+N size out, where M is the window_size
and N is the search_area_size
It leads to inconsistency of the output

openpiv.pyprocess3D.extended_search_area_piv3D(frame_a, frame_b, window_size, overlap=(0, 0, 0), dt=(1.0, 1.0, 1.0), search_area_size=None, correlation_method='fft', subpixel_method='gaussian', sig2noise_method=None, width=2, nfftx=None, nffty=None, nfftz=None)[source]¶

Standard PIV cross-correlation algorithm, with an option for extended area search that increased dynamic range. The search region in the second frame is larger than the interrogation window size in the first frame.

This is a pure python implementation of the standard PIV cross-correlation algorithm. It is a zero order displacement predictor, and no iterative process is performed.

Parameters:

frame_a (3d np.ndarray) – an two dimensions array of integers containing grey levels of the first frame.
frame_b (3d np.ndarray) – an two dimensions array of integers containing grey levels of the second frame.
window_size (tuple) – the size of the (square) interrogation window, [default: 32 pix].
overlap (tuple) – the number of pixels by which two adjacent windows overlap [default: 16 pix].
dt (tuple) – the time delay separating the two frames [default: 1.0].
correlation_method (string) – only one method is currently implemented: ‘fft’
subpixel_method (string) – one of the following methods to estimate subpixel location of the peak: ‘centroid’ [replaces default if correlation map is negative], ‘gaussian’ [default if correlation map is positive], ‘parabolic’.
sig2noise_method (string) – defines the method of signal-to-noise-ratio measure, (‘peak2peak’ or ‘peak2mean’. If None, no measure is performed.)
nfftx (int) – the size of the 3D FFT in x-direction, [default: 2 x windows_a.shape[0] is recommended]
nffty (int) – the size of the 3D FFT in y-direction, [default: 2 x windows_a.shape[1] is recommended]
nfftz (int) – the size of the 3D FFT in z-direction, [default: 2 x windows_a.shape[2] is recommended]
width (int) – the half size of the region around the first correlation peak to ignore for finding the second peak. [default: 2]. Only used if sig2noise_method==peak2peak.
search_area_size (tuple) – the size of the interrogation window in the second frame, default is the same interrogation window size and it is a fallback to the simplest FFT based PIV

Returns:

u (3d np.ndarray) – a three dimensional array containing the u velocity component, in pixels/seconds.
v (3d np.ndarray) – a three dimensional array containing the v velocity component, in pixels/seconds.
w (3d np.ndarray) – a three dimensional array containing the w velocity component, in pixels/seconds.
sig2noise (3d np.ndarray, (optional: only if sig2noise_method is not None)) – a three dimensional array the signal to noise ratio for each window pair.

openpiv.pyprocess3D.find_second_peak_3D(corr, i=None, j=None, z=None, width=2)[source]¶

Find the value of the second largest peak.

The second largest peak is the height of the peak in the region outside a 3x3 submatrix around the first correlation peak.

Parameters:

corr (np.ndarray) – the correlation map.
i,j,z (ints) – row, column and layer location of the first peak.
width (int) – the half size of the region around the first correlation peak to ignore for finding the second peak.

Returns:

i (int) – the row index of the second correlation peak.
j (int) – the column index of the second correlation peak.
z (int) – the 3rd index of the second correlation peak.

corr_max2 : int: the value of the second correlation peak.

openpiv.pyprocess3D.find_subpixel_peak_position(corr, subpixel_method='gaussian')[source]¶

Find subpixel approximation of the correlation peak.

This function returns a subpixels approximation of the correlation peak by using one of the several methods available. If requested, the function also returns the signal to noise ratio level evaluated from the correlation map.

Parameters:	corr (np.ndarray) – the correlation map. subpixel_method (string) – one of the following methods to estimate subpixel location of the peak: ‘centroid’ [replaces default if correlation map is negative], ‘gaussian’ [default if correlation map is positive], ‘parabolic’.
Returns:	subp_peak_position – the fractional row and column indices for the sub-pixel approximation of the correlation peak.
Return type:	two elements tuple

openpiv.pyprocess3D.get_coordinates(image_size, search_area_size, window_size, overlap)[source]¶

Compute the x, y coordinates of the centers of the interrogation windows.

Parameters:

image_size (two elements tuple) – a three dimensional tuple for the pixel size of the image
window_size (tuple) – the size of the interrogation window.
search_area_size (tuple) – the size of the search area window.
overlap (tuple) – the number of pixel by which two adjacent interrogation windows overlap.

Returns:

x (23 np.ndarray) – a three dimensional array containing the x coordinates of the interrogation window centers, in pixels.
y (23 np.ndarray) – a three dimensional array containing the y coordinates of the interrogation window centers, in pixels.
z (23 np.ndarray) – a three dimensional array containing the y coordinates of the interrogation window centers, in pixels.

openpiv.pyprocess3D.nextpower2(i)[source]¶: Find 2^n that is equal to or greater than.

openpiv.pyprocess3D.normalize_intensity(window)[source]¶

Normalize interrogation window by removing the mean value.

Parameters:	window (2d np.ndarray) – the interrogation window array
Returns:	window – the interrogation window array, with mean value equal to zero.
Return type:	2d np.ndarray

openpiv.pyprocess3D.sig2noise_ratio(corr, sig2noise_method='peak2peak', width=2)[source]¶

Computes the signal to noise ratio from the correlation map.

The signal to noise ratio is computed from the correlation map with one of two available method. It is a measure of the quality of the matching between to interogation windows.

Parameters:	corr (2d np.ndarray) – the correlation map. sig2noise_method (string) – the method for evaluating the signal to noise ratio value from the correlation map. Can be peak2peak, peak2mean or None if no evaluation should be made. width (int, optional) – the half size of the region around the first correlation peak to ignore for finding the second peak. [default: 2]. Only used if `sig2noise_method==peak2peak`.
Returns:	sig2noise – the signal to noise ratio from the correlation map.
Return type:	float

openpiv.scaling module¶

Scaling utilities

openpiv.scaling.uniform(x, y, u, v, scaling_factor)[source]¶

Apply an uniform scaling

Parameters:

x (2d np.ndarray) –
y (2d np.ndarray) –
u (2d np.ndarray) –
v (2d np.ndarray) –
scaling_factor (float) – the image scaling factor in pixels per meter

Returns:

x (2d np.ndarray)
y (2d np.ndarray)
u (2d np.ndarray)
v (2d np.ndarray)

openpiv.smoothn module¶

openpiv.smoothn.InitialGuess(y, I)[source]¶

openpiv.smoothn.RobustWeights(r, I, h, wstr)[source]¶

openpiv.smoothn.beta(a, b, size=None)¶

Draw samples from a Beta distribution.

The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function

\[f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1},\]

where the normalization, B, is the beta function,

\[B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt.\]

It is often seen in Bayesian inference and order statistics.

Note

New code should use the ~numpy.random.Generator.beta method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	a (float or array_like of floats) – Alpha, positive (>0). b (float or array_like of floats) – Beta, positive (>0). size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `a` and `b` are both scalars. Otherwise, `np.broadcast(a, b).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized beta distribution.
Return type:	ndarray or scalar

See also

random.Generator.beta(): which should be used for new code.

openpiv.smoothn.binomial(n, p, size=None)¶

Draw samples from a binomial distribution.

Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use)

Note

New code should use the ~numpy.random.Generator.binomial method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	n (int or array_like of ints) – Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers. p (float or array_like of floats) – Parameter of the distribution, >= 0 and <=1. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `n` and `p` are both scalars. Otherwise, `np.broadcast(n, p).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials.
Return type:	ndarray or scalar

See also

scipy.stats.binom(): probability density function, distribution or cumulative density function, etc.
random.Generator.binomial(): which should be used for new code.

Notes

The probability density for the binomial distribution is

\[P(N) = \binom{n}{N}p^N(1-p)^{n-N},\]

where $n$ is the number of trials, $p$ is the probability of success, and $N$ is the number of successes.

When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case.

References

[1]	Dalgaard, Peter, “Introductory Statistics with R”, Springer-Verlag, 2002.

[2]	Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.

[3]	Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.

[4]	Weisstein, Eric W. “Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html

[5]	Wikipedia, “Binomial distribution”, https://en.wikipedia.org/wiki/Binomial_distribution

Examples

Draw samples from the distribution:

>>> n, p = 10, .5  # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.

A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening?

Let’s do 20,000 trials of the model, and count the number that generate zero positive results.

>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
# answer = 0.38885, or 38%.

openpiv.smoothn.chisquare(df, size=None)¶

Draw samples from a chi-square distribution.

When df independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing.

Note

New code should use the ~numpy.random.Generator.chisquare method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	df (float or array_like of floats) – Number of degrees of freedom, must be > 0. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `df` is a scalar. Otherwise, `np.array(df).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized chi-square distribution.
Return type:	ndarray or scalar
Raises:	`ValueError` – When df <= 0 or when an inappropriate size (e.g. `size=-1`) is given.

See also

random.Generator.chisquare(): which should be used for new code.

Notes

The variable obtained by summing the squares of df independent, standard normally distributed random variables:

\[Q = \sum_{i=0}^{\mathtt{df}} X^2_i\]

is chi-square distributed, denoted

\[Q \sim \chi^2_k.\]

The probability density function of the chi-squared distribution is

\[p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2},\]

where $\Gamma$ is the gamma function,

\[\Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.\]

References

[1]	NIST “Engineering Statistics Handbook” https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

Examples

>>> np.random.chisquare(2,4)
array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272]) # random

openpiv.smoothn.choice(a, size=None, replace=True, p=None)¶

Generates a random sample from a given 1-D array

New in version 1.7.0.

Note

New code should use the ~numpy.random.Generator.choice method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	a (1-D array-like or int) – If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if it were `np.arange(a)` size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned. replace (boolean, optional) – Whether the sample is with or without replacement. Default is True, meaning that a value of `a` can be selected multiple times. p (1-D array-like, optional) – The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in `a`.
Returns:	samples – The generated random samples
Return type:	single item or ndarray
Raises:	`ValueError` – If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size

random.Generator.choice(): which should be used in new code

Notes

Setting user-specified probabilities through p uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of p is 1 / len(a).

Sampling random rows from a 2-D array is not possible with this function, but is possible with Generator.choice through its axis keyword.

Examples

Generate a uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3)
array([0, 3, 4]) # random
>>> #This is equivalent to np.random.randint(0,5,3)

Generate a non-uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0]) # random

Generate a uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False)
array([3,1,0]) # random
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]

Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0]) # random

Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance:

>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
      dtype='<U11')

openpiv.smoothn.dctND(data, f=<function dct>)[source]¶

openpiv.smoothn.dirichlet(alpha, size=None)¶

Draw samples from the Dirichlet distribution.

Draw size samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference.

Note

New code should use the ~numpy.random.Generator.dirichlet method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	alpha (sequence of floats, length k) – Parameter of the distribution (length `k` for sample of length `k`). size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n)`, then `m * n * k` samples are drawn. Default is None, in which case a vector of length `k` is returned.
Returns:	samples – The drawn samples, of shape `(size, k)`.
Return type:	ndarray,
Raises:	`ValueError` – If any value in `alpha` is less than or equal to zero

See also

random.Generator.dirichlet(): which should be used for new code.

Notes

The Dirichlet distribution is a distribution over vectors $x$ that fulfil the conditions $x_i>0$ and $\sum_{i=1}^k x_i = 1$.

The probability density function $p$ of a Dirichlet-distributed random vector $X$ is proportional to

\[p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},\]

where $\alpha$ is a vector containing the positive concentration parameters.

The method uses the following property for computation: let $Y$ be a random vector which has components that follow a standard gamma distribution, then $X = \frac{1}{\sum_{i=1}^k{Y_i}} Y$ is Dirichlet-distributed

References

[1]	David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, http://www.inference.org.uk/mackay/itila/

[2]	Wikipedia, “Dirichlet distribution”, https://en.wikipedia.org/wiki/Dirichlet_distribution

Examples

Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces.

>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()

>>> import matplotlib.pyplot as plt
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")

openpiv.smoothn.exponential(scale=1.0, size=None)¶

Draw samples from an exponential distribution.

Its probability density function is

\[f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),\]

for x > 0 and 0 elsewhere. $\beta$ is the scale parameter, which is the inverse of the rate parameter $\lambda = 1/\beta$. The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_.

The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_.

Note

New code should use the ~numpy.random.Generator.exponential method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	scale (float or array_like of floats) – The scale parameter, $\beta = 1/\lambda$. Must be non-negative. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `scale` is a scalar. Otherwise, `np.array(scale).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized exponential distribution.
Return type:	ndarray or scalar

Examples

A real world example: Assume a company has 10000 customer support agents and the average time between customer calls is 4 minutes.

>>> n = 10000
>>> time_between_calls = np.random.default_rng().exponential(scale=4, size=n)

What is the probability that a customer will call in the next 4 to 5 minutes?

>>> x = ((time_between_calls < 5).sum())/n
>>> y = ((time_between_calls < 4).sum())/n
>>> x-y
0.08 # may vary

See also

random.Generator.exponential(): which should be used for new code.

References

[1]	Peyton Z. Peebles Jr., “Probability, Random Variables and Random Signal Principles”, 4th ed, 2001, p. 57.

[2]	Wikipedia, “Poisson process”, https://en.wikipedia.org/wiki/Poisson_process

[3]	Wikipedia, “Exponential distribution”, https://en.wikipedia.org/wiki/Exponential_distribution

openpiv.smoothn.f(dfnum, dfden, size=None)¶

Draw samples from an F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters must be greater than zero.

The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates.

Note

New code should use the ~numpy.random.Generator.f method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	dfnum (float or array_like of floats) – Degrees of freedom in numerator, must be > 0. dfden (float or array_like of float) – Degrees of freedom in denominator, must be > 0. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `dfnum` and `dfden` are both scalars. Otherwise, `np.broadcast(dfnum, dfden).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Fisher distribution.
Return type:	ndarray or scalar

See also

scipy.stats.f(): probability density function, distribution or cumulative density function, etc.
random.Generator.f(): which should be used for new code.

Notes

The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable dfnum is the number of samples minus one, the between-groups degrees of freedom, while dfden is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups.

References

[1]	Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002.

[2]	Wikipedia, “F-distribution”, https://en.wikipedia.org/wiki/F-distribution

Examples

An example from Glantz[1], pp 47-40:

Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children’s blood glucose levels? Calculating the F statistic from the data gives a value of 36.01.

Draw samples from the distribution:

>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)

The lower bound for the top 1% of the samples is :

>>> np.sort(s)[-10]
7.61988120985 # random

So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level.

openpiv.smoothn.gamma(shape, scale=1.0, size=None)¶

Draw samples from a Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale (sometimes designated “theta”), where both parameters are > 0.

Note

New code should use the ~numpy.random.Generator.gamma method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	shape (float or array_like of floats) – The shape of the gamma distribution. Must be non-negative. scale (float or array_like of floats, optional) – The scale of the gamma distribution. Must be non-negative. Default is equal to 1. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `shape` and `scale` are both scalars. Otherwise, `np.broadcast(shape, scale).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized gamma distribution.
Return type:	ndarray or scalar

See also

scipy.stats.gamma(): probability density function, distribution or cumulative density function, etc.
random.Generator.gamma(): which should be used for new code.

Notes

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where $k$ is the shape and $\theta$ the scale, and $\Gamma$ is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

References

[1]	Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html

[2]	Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Examples

Draw samples from the distribution:

>>> shape, scale = 2., 2.  # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  # doctest: +SKIP
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /  # doctest: +SKIP
...                      (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  # doctest: +SKIP
>>> plt.show()

openpiv.smoothn.gcv(p, Lambda, aow, DCTy, IsFinite, Wtot, y, nof, noe, smoothOrder)[source]¶

openpiv.smoothn.geometric(p, size=None)¶

Draw samples from the geometric distribution.

Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, k = 1, 2, ....

The probability mass function of the geometric distribution is

\[f(k) = (1 - p)^{k - 1} p\]

where p is the probability of success of an individual trial.

Note

New code should use the ~numpy.random.Generator.geometric method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	p (float or array_like of floats) – The probability of success of an individual trial. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `p` is a scalar. Otherwise, `np.array(p).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized geometric distribution.
Return type:	ndarray or scalar

See also

random.Generator.geometric(): which should be used for new code.

Examples

Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35:

>>> z = np.random.geometric(p=0.35, size=10000)

How many trials succeeded after a single run?

>>> (z == 1).sum() / 10000.
0.34889999999999999 #random

openpiv.smoothn.get_state(legacy=True)¶

Return a tuple representing the internal state of the generator.

For more details, see set_state.

Parameters: legacy (bool, optional) – Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict. Raises ValueError if the underlying bit generator is not an instance of MT19937.

Returns:

out – If legacy is True, the returned tuple has the following items:

the string ‘MT19937’.
a 1-D array of 624 unsigned integer keys.
an integer pos.
an integer has_gauss.
a float cached_gaussian.

If legacy is False, or the BitGenerator is not MT19937, then state is returned as a dictionary.

Return type: {tuple(str, ndarray of 624 uints, int, int, float), dict}

See also

set_state()

Notes

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

openpiv.smoothn.gumbel(loc=0.0, scale=1.0, size=None)¶

Draw samples from a Gumbel distribution.

Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below.

Note

New code should use the ~numpy.random.Generator.gumbel method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	loc (float or array_like of floats, optional) – The location of the mode of the distribution. Default is 0. scale (float or array_like of floats, optional) – The scale parameter of the distribution. Default is 1. Must be non- negative. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `loc` and `scale` are both scalars. Otherwise, `np.broadcast(loc, scale).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Gumbel distribution.
Return type:	ndarray or scalar

See also

scipy.stats.gumbel_l(), scipy.stats.gumbel_r(), scipy.stats.genextreme(), weibull()

random.Generator.gumbel(): which should be used for new code.

Notes

The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with “exponential-like” tails.

The probability density for the Gumbel distribution is

\[p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}},\]

where $\mu$ is the mode, a location parameter, and $\beta$ is the scale parameter.

The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a “fat-tailed” distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events.

It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet.

The function has a mean of $\mu + 0.57721\beta$ and a variance of $\frac{\pi^2}{6}\beta^2$.

References

[1]	Gumbel, E. J., “Statistics of Extremes,” New York: Columbia University Press, 1958.

[2]	Reiss, R.-D. and Thomas, M., “Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields,” Basel: Birkhauser Verlag, 2001.

Examples

Draw samples from the distribution:

>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp( -np.exp( -(bins - mu) /beta) ),
...          linewidth=2, color='r')
>>> plt.show()

Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:

>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
...    a = np.random.normal(mu, beta, 1000)
...    means.append(a.mean())
...    maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, density=True)
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
>>> mu = np.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp(-np.exp(-(bins - mu)/beta)),
...          linewidth=2, color='r')
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
...          * np.exp(-(bins - mu)**2 / (2 * beta**2)),
...          linewidth=2, color='g')
>>> plt.show()

openpiv.smoothn.hypergeometric(ngood, nbad, nsample, size=None)¶

Draw samples from a Hypergeometric distribution.

Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample (number of items sampled, which is less than or equal to the sum ngood + nbad).

Note

New code should use the ~numpy.random.Generator.hypergeometric method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	ngood (int or array_like of ints) – Number of ways to make a good selection. Must be nonnegative. nbad (int or array_like of ints) – Number of ways to make a bad selection. Must be nonnegative. nsample (int or array_like of ints) – Number of items sampled. Must be at least 1 and at most `ngood + nbad`. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if ngood, nbad, and nsample are all scalars. Otherwise, `np.broadcast(ngood, nbad, nsample).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size nsample taken from a set of ngood good items and nbad bad items.
Return type:	ndarray or scalar

See also

scipy.stats.hypergeom(): probability density function, distribution or cumulative density function, etc.
random.Generator.hypergeometric(): which should be used for new code.

Notes

The probability density for the Hypergeometric distribution is

\[P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},\]

where $0 \le x \le n$ and $n-b \le x \le g$

for P(x) the probability of x good results in the drawn sample, g = ngood, b = nbad, and n = nsample.

Consider an urn with black and white marbles in it, ngood of them are black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample.

Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial.

References

[1]	Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.

[2]	Weisstein, Eric W. “Hypergeometric Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html

[3]	Wikipedia, “Hypergeometric distribution”, https://en.wikipedia.org/wiki/Hypergeometric_distribution

Examples

Draw samples from the distribution:

>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> from matplotlib.pyplot import hist
>>> hist(s)
#   note that it is very unlikely to grab both bad items

Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color?

>>> s = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
#   answer = 0.003 ... pretty unlikely!

openpiv.smoothn.laplace(loc=0.0, scale=1.0, size=None)¶

Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay).

The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables.

Note

New code should use the ~numpy.random.Generator.laplace method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	loc (float or array_like of floats, optional) – The position, $\mu$, of the distribution peak. Default is 0. scale (float or array_like of floats, optional) – $\lambda$, the exponential decay. Default is 1. Must be non- negative. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `loc` and `scale` are both scalars. Otherwise, `np.broadcast(loc, scale).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Laplace distribution.
Return type:	ndarray or scalar

See also

random.Generator.laplace(): which should be used for new code.

Notes

It has the probability density function

\[f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right).\]

The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution.

References

[1]	Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.

[2]	Kotz, Samuel, et. al. “The Laplace Distribution and Generalizations, ” Birkhauser, 2001.

[3]	Weisstein, Eric W. “Laplace Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html

[4]	Wikipedia, “Laplace distribution”, https://en.wikipedia.org/wiki/Laplace_distribution

Examples

Draw samples from the distribution

>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> x = np.arange(-8., 8., .01)
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
>>> plt.plot(x, pdf)

Plot Gaussian for comparison:

>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
...      np.exp(-(x - loc)**2 / (2 * scale**2)))
>>> plt.plot(x,g)

openpiv.smoothn.logistic(loc=0.0, scale=1.0, size=None)¶

Draw samples from a logistic distribution.

Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0).

Note

New code should use the ~numpy.random.Generator.logistic method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	loc (float or array_like of floats, optional) – Parameter of the distribution. Default is 0. scale (float or array_like of floats, optional) – Parameter of the distribution. Must be non-negative. Default is 1. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `loc` and `scale` are both scalars. Otherwise, `np.broadcast(loc, scale).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized logistic distribution.
Return type:	ndarray or scalar

See also

scipy.stats.logistic(): probability density function, distribution or cumulative density function, etc.
random.Generator.logistic(): which should be used for new code.

Notes

The probability density for the Logistic distribution is

\[P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},\]

where $\mu$ = location and $s$ = scale.

The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable.

References

[1]	Reiss, R.-D. and Thomas M. (2001), “Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields,” Birkhauser Verlag, Basel, pp 132-133.

[2]	Weisstein, Eric W. “Logistic Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html

[3]	Wikipedia, “Logistic-distribution”, https://en.wikipedia.org/wiki/Logistic_distribution

Examples

Draw samples from the distribution:

>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=50)

# plot against distribution

>>> def logist(x, loc, scale):
...     return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
>>> lgst_val = logist(bins, loc, scale)
>>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
>>> plt.show()

openpiv.smoothn.lognormal(mean=0.0, sigma=1.0, size=None)¶

Draw samples from a log-normal distribution.

Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from.

Note

New code should use the ~numpy.random.Generator.lognormal method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	mean (float or array_like of floats, optional) – Mean value of the underlying normal distribution. Default is 0. sigma (float or array_like of floats, optional) – Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `mean` and `sigma` are both scalars. Otherwise, `np.broadcast(mean, sigma).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized log-normal distribution.
Return type:	ndarray or scalar

See also

scipy.stats.lognorm(): probability density function, distribution, cumulative density function, etc.
random.Generator.lognormal(): which should be used for new code.

Notes

A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:

\[p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}\]

where $\mu$ is the mean and $\sigma$ is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.

References

[1]	Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

[2]	Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.

Examples

Draw samples from the distribution:

>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')

>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))

>>> plt.plot(x, pdf, linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()

Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.

>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
...    a = 10. + np.random.standard_normal(100)
...    b.append(np.prod(a))

>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
>>> sigma = np.std(np.log(b))
>>> mu = np.mean(np.log(b))

>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))

>>> plt.plot(x, pdf, color='r', linewidth=2)
>>> plt.show()

openpiv.smoothn.logseries(p, size=None)¶

Draw samples from a logarithmic series distribution.

Samples are drawn from a log series distribution with specified shape parameter, 0 <= p < 1.

Note

New code should use the ~numpy.random.Generator.logseries method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	p (float or array_like of floats) – Shape parameter for the distribution. Must be in the range [0, 1). size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `p` is a scalar. Otherwise, `np.array(p).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized logarithmic series distribution.
Return type:	ndarray or scalar

See also

scipy.stats.logser(): probability density function, distribution or cumulative density function, etc.
random.Generator.logseries(): which should be used for new code.

Notes

The probability density for the Log Series distribution is

\[P(k) = \frac{-p^k}{k \ln(1-p)},\]

where p = probability.

The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3].

References

[1]	Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9).

[2]	Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58.

[3]	D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994.

[4]	Wikipedia, “Logarithmic distribution”, https://en.wikipedia.org/wiki/Logarithmic_distribution

Examples

Draw samples from the distribution:

>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s)

# plot against distribution

>>> def logseries(k, p):
...     return -p**k/(k*np.log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/
...          logseries(bins, a).max(), 'r')
>>> plt.show()

openpiv.smoothn.multinomial(n, pvals, size=None)¶

Draw samples from a multinomial distribution.

The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of p possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents n such experiments. Its values, X_i = [X_0, X_1, ..., X_p], represent the number of times the outcome was i.

Note

New code should use the ~numpy.random.Generator.multinomial method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:

n (int) – Number of experiments.
pvals (sequence of floats, length p) – Probabilities of each of the p different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as sum(pvals[:-1]) <= 1).
size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

Returns:

out – The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Return type:

ndarray

See also

random.Generator.multinomial(): which should be used for new code.

Examples

Throw a dice 20 times:

>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]]) # random

It landed 4 times on 1, once on 2, etc.

Now, throw the dice 20 times, and 20 times again:

>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3], # random
       [2, 4, 3, 4, 0, 7]])

For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc.

A loaded die is more likely to land on number 6:

>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26]) # random

The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so:

>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3])  # RIGHT
array([38, 62]) # random

not like:

>>> np.random.multinomial(100, [1.0, 2.0])  # WRONG
Traceback (most recent call last):
ValueError: pvals < 0, pvals > 1 or pvals contains NaNs

openpiv.smoothn.multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)¶

Draw random samples from a multivariate normal distribution.

The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or “center”) and variance (standard deviation, or “width,” squared) of the one-dimensional normal distribution.

Note

New code should use the ~numpy.random.Generator.multivariate_normal method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:

mean (1-D array_like, of length N) – Mean of the N-dimensional distribution.
cov (2-D array_like, of shape (N, N)) – Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling.
size (int or tuple of ints, optional) – Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). If no shape is specified, a single (N-D) sample is returned.
check_valid ({ 'warn', 'raise', 'ignore' }, optional) – Behavior when the covariance matrix is not positive semidefinite.
tol (float, optional) – Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check.

Returns:

out – The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Return type:

ndarray

See also

random.Generator.multivariate_normal(): which should be used for new code.

Notes

The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution.

Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, $X = [x_1, x_2, ... x_N]$. The covariance matrix element $C_{ij}$ is the covariance of $x_i$ and $x_j$. The element $C_{ii}$ is the variance of $x_i$ (i.e. its “spread”).

Instead of specifying the full covariance matrix, popular approximations include:

Spherical covariance (cov is a multiple of the identity matrix)

Diagonal covariance (cov has non-negative elements, and only on the diagonal)

This geometrical property can be seen in two dimensions by plotting generated data-points:

>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]]  # diagonal covariance

Diagonal covariance means that points are oriented along x or y-axis:

>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()

Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.

References

[1]	Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991.

[2]	Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern Classification,” 2nd ed., New York: Wiley, 2001.

Examples

>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)

Here we generate 800 samples from the bivariate normal distribution with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The expected variances of the first and second components of the sample are 6 and 3.5, respectively, and the expected correlation coefficient is -3/sqrt(6*3.5) ≈ -0.65465.

>>> cov = np.array([[6, -3], [-3, 3.5]])
>>> pts = np.random.multivariate_normal([0, 0], cov, size=800)

Check that the mean, covariance, and correlation coefficient of the sample are close to the expected values:

>>> pts.mean(axis=0)
array([ 0.0326911 , -0.01280782])  # may vary
>>> np.cov(pts.T)
array([[ 5.96202397, -2.85602287],
       [-2.85602287,  3.47613949]])  # may vary
>>> np.corrcoef(pts.T)[0, 1]
-0.6273591314603949  # may vary

We can visualize this data with a scatter plot. The orientation of the point cloud illustrates the negative correlation of the components of this sample.

>>> import matplotlib.pyplot as plt
>>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5)
>>> plt.axis('equal')
>>> plt.grid()
>>> plt.show()

openpiv.smoothn.negative_binomial(n, p, size=None)¶

Draw samples from a negative binomial distribution.

Samples are drawn from a negative binomial distribution with specified parameters, n successes and p probability of success where n is > 0 and p is in the interval [0, 1].

Note

New code should use the ~numpy.random.Generator.negative_binomial method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	n (float or array_like of floats) – Parameter of the distribution, > 0. p (float or array_like of floats) – Parameter of the distribution, >= 0 and <=1. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `n` and `p` are both scalars. Otherwise, `np.broadcast(n, p).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached.
Return type:	ndarray or scalar

See also

random.Generator.negative_binomial(): which should be used for new code.

Notes

The probability mass function of the negative binomial distribution is

\[P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},\]

where $n$ is the number of successes, $p$ is the probability of success, $N+n$ is the number of trials, and $\Gamma$ is the gamma function. When $n$ is an integer, $\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}$, which is the more common form of this term in the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial.

If one throws a die repeatedly until the third time a “1” appears, then the probability distribution of the number of non-“1”s that appear before the third “1” is a negative binomial distribution.

References

[1]	Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html

[2]	Wikipedia, “Negative binomial distribution”, https://en.wikipedia.org/wiki/Negative_binomial_distribution

Examples

Draw samples from the distribution:

A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.?

>>> s = np.random.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11): # doctest: +SKIP
...    probability = sum(s<i) / 100000.
...    print(i, "wells drilled, probability of one success =", probability)

openpiv.smoothn.noncentral_chisquare(df, nonc, size=None)¶

Draw samples from a noncentral chi-square distribution.

The noncentral $\chi^2$ distribution is a generalization of the $\chi^2$ distribution.

Note

New code should use the ~numpy.random.Generator.noncentral_chisquare method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	df (float or array_like of floats) – Degrees of freedom, must be > 0. Changed in version 1.10.0: Earlier NumPy versions required dfnum > 1. nonc (float or array_like of floats) – Non-centrality, must be non-negative. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `df` and `nonc` are both scalars. Otherwise, `np.broadcast(df, nonc).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized noncentral chi-square distribution.
Return type:	ndarray or scalar

See also

random.Generator.noncentral_chisquare(): which should be used for new code.

Notes

The probability density function for the noncentral Chi-square distribution is

\[P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x),\]

where $Y_{q}$ is the Chi-square with q degrees of freedom.

References

[1]	Wikipedia, “Noncentral chi-squared distribution” https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

Examples

Draw values from the distribution and plot the histogram

>>> import matplotlib.pyplot as plt
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()

Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
...                   bins=np.arange(0., 25, .1), density=True)
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
...                    bins=np.arange(0., 25, .1), density=True)
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
>>> plt.show()

Demonstrate how large values of non-centrality lead to a more symmetric distribution.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()

openpiv.smoothn.noncentral_f(dfnum, dfden, nonc, size=None)¶

Draw samples from the noncentral F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters > 1. nonc is the non-centrality parameter.

Note

New code should use the ~numpy.random.Generator.noncentral_f method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	dfnum (float or array_like of floats) – Numerator degrees of freedom, must be > 0. Changed in version 1.14.0: Earlier NumPy versions required dfnum > 1. dfden (float or array_like of floats) – Denominator degrees of freedom, must be > 0. nonc (float or array_like of floats) – Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `dfnum`, `dfden`, and `nonc` are all scalars. Otherwise, `np.broadcast(dfnum, dfden, nonc).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized noncentral Fisher distribution.
Return type:	ndarray or scalar

See also

random.Generator.noncentral_f(): which should be used for new code.

Notes

When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic.

References

[1]	Weisstein, Eric W. “Noncentral F-Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html

[2]	Wikipedia, “Noncentral F-distribution”, https://en.wikipedia.org/wiki/Noncentral_F-distribution

Examples

In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We’ll plot the two probability distributions for comparison.

>>> dfnum = 3 # between group deg of freedom
>>> dfden = 20 # within groups degrees of freedom
>>> nonc = 3.0
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
>>> NF = np.histogram(nc_vals, bins=50, density=True)
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
>>> F = np.histogram(c_vals, bins=50, density=True)
>>> import matplotlib.pyplot as plt
>>> plt.plot(F[1][1:], F[0])
>>> plt.plot(NF[1][1:], NF[0])
>>> plt.show()

openpiv.smoothn.normal(loc=0.0, scale=1.0, size=None)¶

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_.

Note

New code should use the ~numpy.random.Generator.normal method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	loc (float or array_like of floats) – Mean (“centre”) of the distribution. scale (float or array_like of floats) – Standard deviation (spread or “width”) of the distribution. Must be non-negative. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `loc` and `scale` are both scalars. Otherwise, `np.broadcast(loc, scale).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized normal distribution.
Return type:	ndarray or scalar

See also

scipy.stats.norm(): probability density function, distribution or cumulative density function, etc.
random.Generator.normal(): which should be used for new code.

Notes

The probability density for the Gaussian distribution is

\[p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },\]

where $\mu$ is the mean and $\sigma$ the standard deviation. The square of the standard deviation, $\sigma^2$, is called the variance.

The function has its peak at the mean, and its “spread” increases with the standard deviation (the function reaches 0.607 times its maximum at $x + \sigma$ and $x - \sigma$ [2]_). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.

References

[1]	Wikipedia, “Normal distribution”, https://en.wikipedia.org/wiki/Normal_distribution

[2]	P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.

Examples

Draw samples from the distribution:

>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)

Verify the mean and the variance:

>>> abs(mu - np.mean(s))
0.0  # may vary

>>> abs(sigma - np.std(s, ddof=1))
0.1  # may vary

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
...          linewidth=2, color='r')
>>> plt.show()

Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5:

>>> np.random.normal(3, 2.5, size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

openpiv.smoothn.pareto(a, size=None)¶

Draw samples from a Pareto II or Lomax distribution with specified shape.

The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter m (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is mu, where the standard Pareto distribution has location mu = 1. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero.

The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the “80-20 rule”. In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range.

Note

New code should use the ~numpy.random.Generator.pareto method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	a (float or array_like of floats) – Shape of the distribution. Must be positive. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `a` is a scalar. Otherwise, `np.array(a).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Pareto distribution.
Return type:	ndarray or scalar

See also

scipy.stats.lomax(): probability density function, distribution or cumulative density function, etc.
scipy.stats.genpareto(): probability density function, distribution or cumulative density function, etc.
random.Generator.pareto(): which should be used for new code.

Notes

The probability density for the Pareto distribution is

\[p(x) = \frac{am^a}{x^{a+1}}\]

where $a$ is the shape and $m$ the scale.

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called “fat-tailed” distributions.

References

[1]	Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects.

[2]	Pareto, V. (1896). Course of Political Economy. Lausanne.

[3]	Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30.

[4]	Wikipedia, “Pareto distribution”, https://en.wikipedia.org/wiki/Pareto_distribution

Examples

Draw samples from the distribution:

>>> a, m = 3., 2.  # shape and mode
>>> s = (np.random.pareto(a, 1000) + 1) * m

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, 100, density=True)
>>> fit = a*m**a / bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
>>> plt.show()

openpiv.smoothn.peaks(n)[source]¶: Mimic basic of matlab peaks fn

openpiv.smoothn.permutation(x)¶

Randomly permute a sequence, or return a permuted range.

If x is a multi-dimensional array, it is only shuffled along its first index.

Note

New code should use the ~numpy.random.Generator.permutation method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	x (int or array_like) – If x is an integer, randomly permute `np.arange(x)`. If x is an array, make a copy and shuffle the elements randomly.
Returns:	out – Permuted sequence or array range.
Return type:	ndarray

See also

random.Generator.permutation(): which should be used for new code.

Examples

>>> np.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random

>>> np.random.permutation([1, 4, 9, 12, 15])
array([15,  1,  9,  4, 12]) # random

>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.permutation(arr)
array([[6, 7, 8], # random
       [0, 1, 2],
       [3, 4, 5]])

openpiv.smoothn.poisson(lam=1.0, size=None)¶

Draw samples from a Poisson distribution.

The Poisson distribution is the limit of the binomial distribution for large N.

Note

New code should use the ~numpy.random.Generator.poisson method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	lam (float or array_like of floats) – Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `lam` is a scalar. Otherwise, `np.array(lam).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Poisson distribution.
Return type:	ndarray or scalar

See also

random.Generator.poisson(): which should be used for new code.

Notes

The Poisson distribution

\[f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}\]

For events with an expected separation $\lambda$ the Poisson distribution $f(k; \lambda)$ describes the probability of $k$ events occurring within the observed interval $\lambda$.

Because the output is limited to the range of the C int64 type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.

References

[1]	Weisstein, Eric W. “Poisson Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html

[2]	Wikipedia, “Poisson distribution”, https://en.wikipedia.org/wiki/Poisson_distribution

Examples

Draw samples from the distribution:

>>> import numpy as np
>>> s = np.random.poisson(5, 10000)

Display histogram of the sample:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 14, density=True)
>>> plt.show()

Draw each 100 values for lambda 100 and 500:

>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))

openpiv.smoothn.power(a, size=None)¶

Draws samples in [0, 1] from a power distribution with positive exponent a - 1.

Also known as the power function distribution.

Note

New code should use the ~numpy.random.Generator.power method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	a (float or array_like of floats) – Parameter of the distribution. Must be non-negative. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `a` is a scalar. Otherwise, `np.array(a).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized power distribution.
Return type:	ndarray or scalar
Raises:	`ValueError` – If a <= 0.

See also

random.Generator.power(): which should be used for new code.

Notes

The probability density function is

\[P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.\]

The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution.

It is used, for example, in modeling the over-reporting of insurance claims.

References

[1]	Christian Kleiber, Samuel Kotz, “Statistical size distributions in economics and actuarial sciences”, Wiley, 2003.

[2]	Heckert, N. A. and Filliben, James J. “NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions”, National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf

Examples

Draw samples from the distribution:

>>> a = 5. # shape
>>> samples = 1000
>>> s = np.random.power(a, samples)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=30)
>>> x = np.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*np.diff(bins)[0]*y
>>> plt.plot(x, normed_y)
>>> plt.show()

Compare the power function distribution to the inverse of the Pareto.

>>> from scipy import stats # doctest: +SKIP
>>> rvs = np.random.power(5, 1000000)
>>> rvsp = np.random.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)  # doctest: +SKIP

>>> plt.figure()
>>> plt.hist(rvs, bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  # doctest: +SKIP
>>> plt.title('np.random.power(5)')

>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  # doctest: +SKIP
>>> plt.title('inverse of 1 + np.random.pareto(5)')

>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  # doctest: +SKIP
>>> plt.title('inverse of stats.pareto(5)')

openpiv.smoothn.rand(d0, d1, ..., dn)¶

Random values in a given shape.

Note

This is a convenience function for users porting code from Matlab, and wraps random_sample. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Create an array of the given shape and populate it with random samples from a uniform distribution over [0, 1).

Parameters:	d1, .., dn (d0,) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
Returns:	out – Random values.
Return type:	ndarray, shape `(d0, d1, ..., dn)`

See also

random()

Examples

>>> np.random.rand(3,2)
array([[ 0.14022471,  0.96360618],  #random
       [ 0.37601032,  0.25528411],  #random
       [ 0.49313049,  0.94909878]]) #random

openpiv.smoothn.randint(low, high=None, size=None, dtype=int)¶

Return random integers from low (inclusive) to high (exclusive).

Return random integers from the “discrete uniform” distribution of the specified dtype in the “half-open” interval [low, high). If high is None (the default), then results are from [0, low).

Note

New code should use the ~numpy.random.Generator.integers method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	low (int or array-like of ints) – Lowest (signed) integers to be drawn from the distribution (unless `high=None`, in which case this parameter is one above the highest such integer). high (int or array-like of ints, optional) – If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if `high=None`). If array-like, must contain integer values size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned. dtype (dtype, optional) – Desired dtype of the result. Byteorder must be native. The default value is int. New in version 1.11.0.
Returns:	out – size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
Return type:	int or ndarray of ints

See also

random_integers(): similar to randint, only for the closed interval [low, high], and 1 is the lowest value if high is omitted.
random.Generator.integers(): which should be used for new code.

Examples

>>> np.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random
>>> np.random.randint(1, size=10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

Generate a 2 x 4 array of ints between 0 and 4, inclusive:

>>> np.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1], # random
       [3, 2, 2, 0]])

Generate a 1 x 3 array with 3 different upper bounds

>>> np.random.randint(1, [3, 5, 10])
array([2, 2, 9]) # random

Generate a 1 by 3 array with 3 different lower bounds

>>> np.random.randint([1, 5, 7], 10)
array([9, 8, 7]) # random

Generate a 2 by 4 array using broadcasting with dtype of uint8

>>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)
array([[ 8,  6,  9,  7], # random
       [ 1, 16,  9, 12]], dtype=uint8)

openpiv.smoothn.randn(d0, d1, ..., dn)¶

Return a sample (or samples) from the “standard normal” distribution.

Note

This is a convenience function for users porting code from Matlab, and wraps standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Note

New code should use the ~numpy.random.Generator.standard_normal method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

If positive int_like arguments are provided, randn generates an array of shape (d0, d1, ..., dn), filled with random floats sampled from a univariate “normal” (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided.

Parameters:	d1, .., dn (d0,) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.
Returns:	Z – A `(d0, d1, ..., dn)`-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied.
Return type:	ndarray or float

See also

standard_normal(): Similar, but takes a tuple as its argument.
normal(): Also accepts mu and sigma arguments.
random.Generator.standard_normal(): which should be used for new code.

Notes

For random samples from the normal distribution with mean mu and standard deviation sigma, use:

sigma * np.random.randn(...) + mu

Examples

>>> np.random.randn()
2.1923875335537315  # random

Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5:

>>> 3 + 2.5 * np.random.randn(2, 4)
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

openpiv.smoothn.random(size=None)¶: Return random floats in the half-open interval [0.0, 1.0). Alias for random_sample to ease forward-porting to the new random API.

openpiv.smoothn.random_integers(low, high=None, size=None)¶

Random integers of type np.int_ between low and high, inclusive.

Return random integers of type np.int_ from the “discrete uniform” distribution in the closed interval [low, high]. If high is None (the default), then results are from [1, low]. The np.int_ type translates to the C long integer type and its precision is platform dependent.

This function has been deprecated. Use randint instead.

Deprecated since version 1.11.0.

Parameters:	low (int) – Lowest (signed) integer to be drawn from the distribution (unless `high=None`, in which case this parameter is the highest such integer). high (int, optional) – If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if `high=None`). size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned.
Returns:	out – size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.
Return type:	int or ndarray of ints

See also

randint(): Similar to random_integers, only for the half-open interval [low, high), and 0 is the lowest value if high is omitted.

Notes

To sample from N evenly spaced floating-point numbers between a and b, use:

a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)

Examples

>>> np.random.random_integers(5)
4 # random
>>> type(np.random.random_integers(5))
<class 'numpy.int64'>
>>> np.random.random_integers(5, size=(3,2))
array([[5, 4], # random
       [3, 3],
       [4, 5]])

Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (i.e., from the set ${0, 5/8, 10/8, 15/8, 20/8}$):

>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625,  1.25 ,  0.625,  0.625,  2.5  ]) # random

Roll two six sided dice 1000 times and sum the results:

>>> d1 = np.random.random_integers(1, 6, 1000)
>>> d2 = np.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2

Display results as a histogram:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, density=True)
>>> plt.show()

openpiv.smoothn.random_sample(size=None)¶

Return random floats in the half-open interval [0.0, 1.0).

Results are from the “continuous uniform” distribution over the stated interval. To sample $Unif[a, b), b > a$ multiply the output of random_sample by (b-a) and add a:

(b - a) * random_sample() + a

Note

New code should use the ~numpy.random.Generator.random method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned.
Returns:	out – Array of random floats of shape size (unless `size=None`, in which case a single float is returned).
Return type:	float or ndarray of floats

See also

random.Generator.random(): which should be used for new code.

Examples

>>> np.random.random_sample()
0.47108547995356098 # random
>>> type(np.random.random_sample())
<class 'float'>
>>> np.random.random_sample((5,))
array([ 0.30220482,  0.86820401,  0.1654503 ,  0.11659149,  0.54323428]) # random

Three-by-two array of random numbers from [-5, 0):

>>> 5 * np.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984], # random
       [-2.99091858, -0.79479508],
       [-1.23204345, -1.75224494]])

openpiv.smoothn.rayleigh(scale=1.0, size=None)¶

Draw samples from a Rayleigh distribution.

The $\chi$ and Weibull distributions are generalizations of the Rayleigh.

Note

New code should use the ~numpy.random.Generator.rayleigh method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	scale (float or array_like of floats, optional) – Scale, also equals the mode. Must be non-negative. Default is 1. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `scale` is a scalar. Otherwise, `np.array(scale).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Rayleigh distribution.
Return type:	ndarray or scalar

See also

random.Generator.rayleigh(): which should be used for new code.

Notes

The probability density function for the Rayleigh distribution is

\[P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}\]

The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution.

References

[1]	Brighton Webs Ltd., “Rayleigh Distribution,” https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp

[2]	Wikipedia, “Rayleigh distribution” https://en.wikipedia.org/wiki/Rayleigh_distribution

Examples

Draw values from the distribution and plot the histogram

>>> from matplotlib.pyplot import hist
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True)

Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters?

>>> meanvalue = 1
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
>>> s = np.random.rayleigh(modevalue, 1000000)

The percentage of waves larger than 3 meters is:

>>> 100.*sum(s>3)/1000000.
0.087300000000000003 # random

openpiv.smoothn.set_state(state)¶

Set the internal state of the generator from a tuple.

For use if one has reason to manually (re-)set the internal state of the bit generator used by the RandomState instance. By default, RandomState uses the “Mersenne Twister”[1]_ pseudo-random number generating algorithm.

Parameters:

state ({tuple(str, ndarray of 624 uints, int, int, float), dict}) –

The state tuple has the following items:

the string ‘MT19937’, specifying the Mersenne Twister algorithm.
a 1-D array of 624 unsigned integers keys.
an integer pos.
an integer has_gauss.
a float cached_gaussian.

If state is a dictionary, it is directly set using the BitGenerators state property.

Returns: out – Returns ‘None’ on success.

Return type: None

See also

get_state()

Notes

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: state = ('MT19937', keys, pos).

References

[1]	M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. on Modeling and Computer Simulation, Vol. 8, No. 1, pp. 3-30, Jan. 1998.

openpiv.smoothn.shuffle(x)¶

Modify a sequence in-place by shuffling its contents.

This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.

Note

New code should use the ~numpy.random.Generator.shuffle method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	x (ndarray or MutableSequence) – The array, list or mutable sequence to be shuffled.
Returns:
Return type:	None

See also

random.Generator.shuffle(): which should be used for new code.

Examples

>>> arr = np.arange(10)
>>> np.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8] # random

Multi-dimensional arrays are only shuffled along the first axis:

>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.shuffle(arr)
>>> arr
array([[3, 4, 5], # random
       [6, 7, 8],
       [0, 1, 2]])

openpiv.smoothn.smooth(u, mask)[source]¶

openpiv.smoothn.smooth_masked_array(u)[source]¶: Use smooth() on the masked array

openpiv.smoothn.smoothn(y, nS0=10, axis=None, smoothOrder=2.0, sd=None, verbose=False, s0=None, z0=None, isrobust=False, W=None, s=None, MaxIter=100, TolZ=0.001, weightstr='bisquare')[source]¶

function [z,s,exitflag,Wtot] = smoothn(varargin) SMOOTHN Robust spline smoothing for 1-D to N-D data. SMOOTHN provides a fast, automatized and robust discretized smoothing spline for data of any dimension. Z = SMOOTHN(Y) automatically smoothes the uniformly-sampled array Y. Y can be any N-D noisy array (time series, images, 3D data,…). Non finite data (NaN or Inf) are treated as missing values. Z = SMOOTHN(Y,S) smoothes the array Y using the smoothing parameter S. S must be a real positive scalar. The larger S is, the smoother the output will be. If the smoothing parameter S is omitted (see previous option) or empty (i.e. S = []), it is automatically determined using the generalized cross-validation (GCV) method. Z = SMOOTHN(Y,W) or Z = SMOOTHN(Y,W,S) specifies a weighting array W of real positive values, that must have the same size as Y. Note that a nil weight corresponds to a missing value. Robust smoothing —————- Z = SMOOTHN(…,’robust’) carries out a robust smoothing that minimizes the influence of outlying data. [Z,S] = SMOOTHN(…) also returns the calculated value for S so that you can fine-tune the smoothing subsequently if needed. An iteration process is used in the presence of weighted and/or missing values. Z = SMOOTHN(…,OPTION_NAME,OPTION_VALUE) smoothes with the termination parameters specified by OPTION_NAME and OPTION_VALUE. They can contain the following criteria:

TolZ: Termination tolerance on Z (default = 1e-3)

TolZ must be in ]0,1[

MaxIter: Maximum number of iterations allowed (default = 100) Initial: Initial value for the iterative process (default =

original data)

Syntax: [Z,…] = SMOOTHN(…,’MaxIter’,500,’TolZ’,1e-4,’Initial’,Z0); [Z,S,EXITFLAG] = SMOOTHN(…) returns a boolean value EXITFLAG that describes the exit condition of SMOOTHN:

1 SMOOTHN converged. 0 Maximum number of iterations was reached.

Input array can be numeric or logical. The returned array is of class double. Notes —– The N-D (inverse) discrete cosine transform functions <a href=”matlab:web(‘http://www.biomecardio.com/matlab/dctn.html’)” >DCTN</a> and <a href=”matlab:web(‘http://www.biomecardio.com/matlab/idctn.html’)” >IDCTN</a> are required. To be made ———- Estimate the confidence bands (see Wahba 1983, Nychka 1988). Reference ——— Garcia D, Robust smoothing of gridded data in one and higher dimensions with missing values. Computational Statistics & Data Analysis, 2010. <a href=”matlab:web(‘http://www.biomecardio.com/pageshtm/publi/csda10.pdf’)”>PDF download</a> Examples: ——– # 1-D example x = linspace(0,100,2**8); y = cos(x/10)+(x/50)**2 + randn(size(x))/10; y[[70, 75, 80]] = [5.5, 5, 6]; z = smoothn(y); # Regular smoothing zr = smoothn(y,’robust’); # Robust smoothing subplot(121), plot(x,y,’r.’,x,z,’k’,’LineWidth’,2) axis square, title(‘Regular smoothing’) subplot(122), plot(x,y,’r.’,x,zr,’k’,’LineWidth’,2) axis square, title(‘Robust smoothing’) # 2-D example xp = 0:.02:1; [x,y] = meshgrid(xp); f = exp(x+y) + sin((x-2*y)*3); fn = f + randn(size(f))*0.5; fs = smoothn(fn); subplot(121), surf(xp,xp,fn), zlim([0 8]), axis square subplot(122), surf(xp,xp,fs), zlim([0 8]), axis square # 2-D example with missing data n = 256; y0 = peaks(n); y = y0 + rand(size(y0))*2; I = randperm(n^2); y(I(1:n^2*0.5)) = NaN; # lose 1/2 of data y(40:90,140:190) = NaN; # create a hole z = smoothn(y); # smooth data subplot(2,2,1:2), imagesc(y), axis equal off title(‘Noisy corrupt data’) subplot(223), imagesc(z), axis equal off title(‘Recovered data …’) subplot(224), imagesc(y0), axis equal off title(’… compared with original data’) # 3-D example [x,y,z] = meshgrid(-2:.2:2); xslice = [-0.8,1]; yslice = 2; zslice = [-2,0]; vn = x.*exp(-x.^2-y.^2-z.^2) + randn(size(x))*0.06; subplot(121), slice(x,y,z,vn,xslice,yslice,zslice,’cubic’) title(‘Noisy data’) v = smoothn(vn); subplot(122), slice(x,y,z,v,xslice,yslice,zslice,’cubic’) title(‘Smoothed data’) # Cardioid t = linspace(0,2*pi,1000); x = 2*cos(t).*(1-cos(t)) + randn(size(t))*0.1; y = 2*sin(t).*(1-cos(t)) + randn(size(t))*0.1; z = smoothn(complex(x,y)); plot(x,y,’r.’,real(z),imag(z),’k’,’linewidth’,2) axis equal tight # Cellular vortical flow [x,y] = meshgrid(linspace(0,1,24)); Vx = cos(2*pi*x+pi/2).*cos(2*pi*y); Vy = sin(2*pi*x+pi/2).*sin(2*pi*y); Vx = Vx + sqrt(0.05)*randn(24,24); # adding Gaussian noise Vy = Vy + sqrt(0.05)*randn(24,24); # adding Gaussian noise I = randperm(numel(Vx)); Vx(I(1:30)) = (rand(30,1)-0.5)*5; # adding outliers Vy(I(1:30)) = (rand(30,1)-0.5)*5; # adding outliers Vx(I(31:60)) = NaN; # missing values Vy(I(31:60)) = NaN; # missing values Vs = smoothn(complex(Vx,Vy),’robust’); # automatic smoothing subplot(121), quiver(x,y,Vx,Vy,2.5), axis square title(‘Noisy velocity field’) subplot(122), quiver(x,y,real(Vs),imag(Vs)), axis square title(‘Smoothed velocity field’) See also SMOOTH, SMOOTH3, DCTN, IDCTN. – Damien Garcia – 2009/03, revised 2010/11 Visit my <a href=”matlab:web(‘http://www.biomecardio.com/matlab/smoothn.html’)”>website</a> for more details about SMOOTHN

# Check input arguments error(nargchk(1,12,nargin)); z0=None,W=None,s=None,MaxIter=100,TolZ=1e-3

openpiv.smoothn.sparseSVD(D)[source]¶

openpiv.smoothn.sparseTest(n=1000)[source]¶

openpiv.smoothn.standard_cauchy(size=None)¶

Draw samples from a standard Cauchy distribution with mode = 0.

Also known as the Lorentz distribution.

Note

New code should use the ~numpy.random.Generator.standard_cauchy method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned.
Returns:	samples – The drawn samples.
Return type:	ndarray or scalar

See also

random.Generator.standard_cauchy(): which should be used for new code.

Notes

The probability density function for the full Cauchy distribution is

\[P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] }\]

and the Standard Cauchy distribution just sets $x_0=0$ and $\gamma=1$

The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis.

When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails.

References

[1]	NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy Distribution”, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm

[2]	Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html

[3]	Wikipedia, “Cauchy distribution” https://en.wikipedia.org/wiki/Cauchy_distribution

Examples

Draw samples and plot the distribution:

>>> import matplotlib.pyplot as plt
>>> s = np.random.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()

openpiv.smoothn.standard_exponential(size=None)¶

Draw samples from the standard exponential distribution.

standard_exponential is identical to the exponential distribution with a scale parameter of 1.

Note

New code should use the ~numpy.random.Generator.standard_exponential method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned.
Returns:	out – Drawn samples.
Return type:	float or ndarray

See also

random.Generator.standard_exponential(): which should be used for new code.

Examples

Output a 3x8000 array:

>>> n = np.random.standard_exponential((3, 8000))

openpiv.smoothn.standard_gamma(shape, size=None)¶

Draw samples from a standard Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale=1.

Note

New code should use the ~numpy.random.Generator.standard_gamma method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	shape (float or array_like of floats) – Parameter, must be non-negative. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `shape` is a scalar. Otherwise, `np.array(shape).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized standard gamma distribution.
Return type:	ndarray or scalar

See also

scipy.stats.gamma(): probability density function, distribution or cumulative density function, etc.
random.Generator.standard_gamma(): which should be used for new code.

Notes

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where $k$ is the shape and $\theta$ the scale, and $\Gamma$ is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

References

[1]	Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html

[2]	Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Examples

Draw samples from the distribution:

>>> shape, scale = 2., 1. # mean and width
>>> s = np.random.standard_gamma(shape, 1000000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  # doctest: +SKIP
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/  # doctest: +SKIP
...                       (sps.gamma(shape) * scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  # doctest: +SKIP
>>> plt.show()

openpiv.smoothn.standard_normal(size=None)¶

Draw samples from a standard Normal distribution (mean=0, stdev=1).

Note

New code should use the ~numpy.random.Generator.standard_normal method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned.
Returns:	out – A floating-point array of shape `size` of drawn samples, or a single sample if `size` was not specified.
Return type:	float or ndarray

See also

normal(): Equivalent function with additional loc and scale arguments for setting the mean and standard deviation.
random.Generator.standard_normal(): which should be used for new code.

Notes

For random samples from the normal distribution with mean mu and standard deviation sigma, use one of:

mu + sigma * np.random.standard_normal(size=...)
np.random.normal(mu, sigma, size=...)

Examples

>>> np.random.standard_normal()
2.1923875335537315 #random

>>> s = np.random.standard_normal(8000)
>>> s
array([ 0.6888893 ,  0.78096262, -0.89086505, ...,  0.49876311,  # random
       -0.38672696, -0.4685006 ])                                # random
>>> s.shape
(8000,)
>>> s = np.random.standard_normal(size=(3, 4, 2))
>>> s.shape
(3, 4, 2)

Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5:

>>> 3 + 2.5 * np.random.standard_normal(size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

openpiv.smoothn.standard_t(df, size=None)¶

Draw samples from a standard Student’s t distribution with df degrees of freedom.

A special case of the hyperbolic distribution. As df gets large, the result resembles that of the standard normal distribution (standard_normal).

Note

New code should use the ~numpy.random.Generator.standard_t method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	df (float or array_like of floats) – Degrees of freedom, must be > 0. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `df` is a scalar. Otherwise, `np.array(df).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized standard Student’s t distribution.
Return type:	ndarray or scalar

See also

random.Generator.standard_t(): which should be used for new code.

Notes

The probability density function for the t distribution is

\[P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}\]

The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean.

The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student.

References

[1]	Dalgaard, Peter, “Introductory Statistics With R”, Springer, 2002.

[2]	Wikipedia, “Student’s t-distribution” https://en.wikipedia.org/wiki/Student’s_t-distribution

Examples

From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is:

>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
...                    7515, 8230, 8770])

Does their energy intake deviate systematically from the recommended value of 7725 kJ? Our null hypothesis will be the absence of deviation, and the alternate hypothesis will be the presence of an effect that could be either positive or negative, hence making our test 2-tailed.

Because we are estimating the mean and we have N=11 values in our sample, we have N-1=10 degrees of freedom. We set our significance level to 95% and compute the t statistic using the empirical mean and empirical standard deviation of our intake. We use a ddof of 1 to base the computation of our empirical standard deviation on an unbiased estimate of the variance (note: the final estimate is not unbiased due to the concave nature of the square root).

>>> np.mean(intake)
6753.636363636364
>>> intake.std(ddof=1)
1142.1232221373727
>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
>>> t
-2.8207540608310198

We draw 1000000 samples from Student’s t distribution with the adequate degrees of freedom.

>>> import matplotlib.pyplot as plt
>>> s = np.random.standard_t(10, size=1000000)
>>> h = plt.hist(s, bins=100, density=True)

Does our t statistic land in one of the two critical regions found at both tails of the distribution?

>>> np.sum(np.abs(t) < np.abs(s)) / float(len(s))
0.018318  #random < 0.05, statistic is in critical region

The probability value for this 2-tailed test is about 1.83%, which is lower than the 5% pre-determined significance threshold.

Therefore, the probability of observing values as extreme as our intake conditionally on the null hypothesis being true is too low, and we reject the null hypothesis of no deviation.

openpiv.smoothn.test1()[source]¶

openpiv.smoothn.test2(axis=None)[source]¶

openpiv.smoothn.test3(axis=None)[source]¶

openpiv.smoothn.test4(i=10, step=0.2, axis=None)[source]¶

openpiv.smoothn.test5()[source]¶

openpiv.smoothn.test6(noise=0.05, nout=30)[source]¶

openpiv.smoothn.triangular(left, mode, right, size=None)¶

Draw samples from the triangular distribution over the interval [left, right].

The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf.

Note

New code should use the ~numpy.random.Generator.triangular method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	left (float or array_like of floats) – Lower limit. mode (float or array_like of floats) – The value where the peak of the distribution occurs. The value must fulfill the condition `left <= mode <= right`. right (float or array_like of floats) – Upper limit, must be larger than left. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `left`, `mode`, and `right` are all scalars. Otherwise, `np.broadcast(left, mode, right).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized triangular distribution.
Return type:	ndarray or scalar

See also

random.Generator.triangular(): which should be used for new code.

Notes

The probability density function for the triangular distribution is

\[\begin{split}P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases}\end{split}\]

The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations.

References

[1]	Wikipedia, “Triangular distribution” https://en.wikipedia.org/wiki/Triangular_distribution

Examples

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
...              density=True)
>>> plt.show()

openpiv.smoothn.uniform(low=0.0, high=1.0, size=None)¶

Draw samples from a uniform distribution.

Samples are uniformly distributed over the half-open interval [low, high) (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by uniform.

Note

New code should use the ~numpy.random.Generator.uniform method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	low (float or array_like of floats, optional) – Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0. high (float or array_like of floats) – Upper boundary of the output interval. All values generated will be less than or equal to high. The high limit may be included in the returned array of floats due to floating-point rounding in the equation `low + (high-low) * random_sample()`. The default value is 1.0. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `low` and `high` are both scalars. Otherwise, `np.broadcast(low, high).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized uniform distribution.
Return type:	ndarray or scalar

See also

randint(): Discrete uniform distribution, yielding integers.
random_integers(): Discrete uniform distribution over the closed interval [low, high].
random_sample(): Floats uniformly distributed over [0, 1).
random(): Alias for random_sample.
rand(): Convenience function that accepts dimensions as input, e.g., rand(2,2) would generate a 2-by-2 array of floats, uniformly distributed over [0, 1).
random.Generator.uniform(): which should be used for new code.

Notes

The probability density function of the uniform distribution is

\[p(x) = \frac{1}{b - a}\]

anywhere within the interval [a, b), and zero elsewhere.

When high == low, values of low will be returned. If high < low, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. The high limit may be included in the returned array of floats due to floating-point rounding in the equation low + (high-low) * random_sample(). For example:

>>> x = np.float32(5*0.99999999)
>>> x
5.0

Examples

Draw samples from the distribution:

>>> s = np.random.uniform(-1,0,1000)

All values are within the given interval:

>>> np.all(s >= -1)
True
>>> np.all(s < 0)
True

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 15, density=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()

openpiv.smoothn.vonmises(mu, kappa, size=None)¶

Draw samples from a von Mises distribution.

Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi].

The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution.

Note

New code should use the ~numpy.random.Generator.vonmises method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	mu (float or array_like of floats) – Mode (“center”) of the distribution. kappa (float or array_like of floats) – Dispersion of the distribution, has to be >=0. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `mu` and `kappa` are both scalars. Otherwise, `np.broadcast(mu, kappa).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized von Mises distribution.
Return type:	ndarray or scalar

See also

scipy.stats.vonmises(): probability density function, distribution, or cumulative density function, etc.
random.Generator.vonmises(): which should be used for new code.

Notes

The probability density for the von Mises distribution is

\[p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},\]

where $\mu$ is the mode and $\kappa$ the dispersion, and $I_0(\kappa)$ is the modified Bessel function of order 0.

The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science.

References

[1]	Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.

[2]	von Mises, R., “Mathematical Theory of Probability and Statistics”, New York: Academic Press, 1964.

Examples

Draw samples from the distribution:

>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.vonmises(mu, kappa, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import i0  # doctest: +SKIP
>>> plt.hist(s, 50, density=True)
>>> x = np.linspace(-np.pi, np.pi, num=51)
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))  # doctest: +SKIP
>>> plt.plot(x, y, linewidth=2, color='r')  # doctest: +SKIP
>>> plt.show()

openpiv.smoothn.wald(mean, scale, size=None)¶

Draw samples from a Wald, or inverse Gaussian, distribution.

As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal.

The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time.

Note

New code should use the ~numpy.random.Generator.wald method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	mean (float or array_like of floats) – Distribution mean, must be > 0. scale (float or array_like of floats) – Scale parameter, must be > 0. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `mean` and `scale` are both scalars. Otherwise, `np.broadcast(mean, scale).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Wald distribution.
Return type:	ndarray or scalar

See also

random.Generator.wald(): which should be used for new code.

Notes

The probability density function for the Wald distribution is

\[P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x}\]

As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes.

References

[1]	Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp

[2]	Chhikara, Raj S., and Folks, J. Leroy, “The Inverse Gaussian Distribution: Theory : Methodology, and Applications”, CRC Press, 1988.

[3]	Wikipedia, “Inverse Gaussian distribution” https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

Examples

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True)
>>> plt.show()

openpiv.smoothn.warning(s1, s2)[source]¶

openpiv.smoothn.weibull(a, size=None)¶

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.

\[X = (-ln(U))^{1/a}\]

Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter $\lambda$ is just $X = \lambda(-ln(U))^{1/a}$.

Note

New code should use the ~numpy.random.Generator.weibull method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	a (float or array_like of floats) – Shape parameter of the distribution. Must be nonnegative. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `a` is a scalar. Otherwise, `np.array(a).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Weibull distribution.
Return type:	ndarray or scalar

See also

scipy.stats.weibull_max(), scipy.stats.weibull_min(), scipy.stats.genextreme(), gumbel()

random.Generator.weibull(): which should be used for new code.

Notes

The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions.

The probability density for the Weibull distribution is

\[p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},\]

where $a$ is the shape and $\lambda$ the scale.

The function has its peak (the mode) at $\lambda(\frac{a-1}{a})^{1/a}$.

When a = 1, the Weibull distribution reduces to the exponential distribution.

References

[1]	Waloddi Weibull, Royal Technical University, Stockholm, 1939 “A Statistical Theory Of The Strength Of Materials”, Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.

[2]	Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951.

[3]	Wikipedia, “Weibull distribution”, https://en.wikipedia.org/wiki/Weibull_distribution

Examples

Draw samples from the distribution:

>>> a = 5. # shape
>>> s = np.random.weibull(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)

>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()

openpiv.smoothn.zipf(a, size=None)¶

Draw samples from a Zipf distribution.

Samples are drawn from a Zipf distribution with specified parameter a > 1.

The Zipf distribution (also known as the zeta distribution) is a discrete probability distribution that satisfies Zipf’s law: the frequency of an item is inversely proportional to its rank in a frequency table.

Note

New code should use the ~numpy.random.Generator.zipf method of a ~numpy.random.Generator instance instead; please see the random-quick-start.

Parameters:	a (float or array_like of floats) – Distribution parameter. Must be greater than 1. size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `a` is a scalar. Otherwise, `np.array(a).size` samples are drawn.
Returns:	out – Drawn samples from the parameterized Zipf distribution.
Return type:	ndarray or scalar

See also

scipy.stats.zipf(): probability density function, distribution, or cumulative density function, etc.
random.Generator.zipf(): which should be used for new code.

Notes

The probability density for the Zipf distribution is

\[p(k) = \frac{k^{-a}}{\zeta(a)},\]

for integers $k \geq 1$, where $\zeta$ is the Riemann Zeta function.

It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table.

References

[1]	Zipf, G. K., “Selected Studies of the Principle of Relative Frequency in Language,” Cambridge, MA: Harvard Univ. Press, 1932.

Examples

Draw samples from the distribution:

>>> a = 4.0
>>> n = 20000
>>> s = np.random.zipf(a, n)

Display the histogram of the samples, along with the expected histogram based on the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import zeta  # doctest: +SKIP

bincount provides a fast histogram for small integers.

>>> count = np.bincount(s)
>>> k = np.arange(1, s.max() + 1)

>>> plt.bar(k, count[1:], alpha=0.5, label='sample count')
>>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5,
...          label='expected count')   # doctest: +SKIP
>>> plt.semilogy()
>>> plt.grid(alpha=0.4)
>>> plt.legend()
>>> plt.title(f'Zipf sample, a={a}, size={n}')
>>> plt.show()

openpiv.tools module¶

The openpiv.tools module is a collection of utilities and tools.

class openpiv.tools.Multiprocesser(data_dir: pathlib.Path, pattern_a: str, pattern_b: Optional[str] = None)[source]¶

Bases: object

run(func, n_cpus=1)[source]¶

Start to process images.

Parameters:	func (python function which will be executed for each) – image pair. See tutorial for more details. n_cpus (int) – the number of processes to launch in parallel. For debugging purposes use n_cpus=1

openpiv.tools.convert_16bits_tif(filename, save_name)[source]¶

convert 16 bits TIFF to an openpiv readable image

Parameters:	filename (_type_) – filename of a 16 bit TIFF save_name (_type_) – new image filename

openpiv.tools.display(message)[source]¶

Display a message to standard output.

Parameters:	message (string) – a message to be printed

openpiv.tools.display_vector_field(filename: Union[pathlib.Path, str], on_img: Optional[bool] = False, image_name: Union[pathlib.Path, str, None] = None, window_size: Optional[int] = 32, scaling_factor: Optional[float] = 1.0, ax: Optional[Any] = None, width: Optional[float] = 0.0025, show_invalid: Optional[bool] = True, **kw)[source]¶

Displays quiver plot of the data stored in the file

Parameters:

filename (string) – the absolute path of the text file
on_img (Bool, optional) – if True, display the vector field on top of the image provided by image_name
image_name (string, optional) – path to the image to plot the vector field onto when on_img is True
window_size (int, optional) – when on_img is True, provide the interrogation window size to fit the background image to the vector field
scaling_factor (float, optional) – when on_img is True, provide the scaling factor to scale the background image to the vector field
show_invalid (bool, show or not the invalid vectors, default is True) –

Key arguments : (additional parameters, optional): scale: [None | float] width: [None | float]

matplotlib.pyplot.quiver

Examples

— only vector field >>> openpiv.tools.display_vector_field(‘./exp1_0000.txt’,scale=100,

width=0.0025)

— vector field on top of image >>> openpiv.tools.display_vector_field(Path(‘./exp1_0000.txt’), on_img=True,

image_name=Path(‘exp1_001_a.bmp’), window_size=32, scaling_factor=70, scale=100, width=0.0025)

openpiv.tools.display_windows_sampling(x, y, window_size, skip=0, method='standard')[source]¶

Displays a map of the interrogation points and windows

Parameters:

x (2d np.ndarray) – a two dimensional array containing the x coordinates of the interrogation window centers, in pixels.
y (2d np.ndarray) – a two dimensional array containing the y coordinates of the interrogation window centers, in pixels.
window_size (the interrogation window size, in pixels) –
skip (the number of windows to skip on a row during display.) – Recommended value is 0 or 1 for standard method, can be more for random method -1 to not show any window
method (can be only <standard> (uniform sampling and constant window size)) – <random> (pick randomly some windows)

Examples

>>> openpiv.tools.display_windows_sampling(x, y, window_size=32, skip=0, method='standard')

openpiv.tools.edges(list_img, filename)[source]¶

openpiv.tools.find_boundaries(threshold, list_img1, list_img2, filename, picname)[source]¶

openpiv.tools.find_reflexions(list_img, filename)[source]¶

openpiv.tools.imread(filename, flatten=0)[source]¶

Read an image file into a numpy array using imageio imread

Parameters:	filename (string) – the absolute path of the image file flatten (bool) – True if the image is RGB color or False (default) if greyscale
Returns:	frame – a numpy array with grey levels
Return type:	np.ndarray

Examples

>>> image = openpiv.tools.imread( 'image.bmp' )
>>> print image.shape
    (1280, 1024)

openpiv.tools.imsave(filename, arr)[source]¶

Write an image file from a numpy array using imageio imread

Parameters:	filename (string) – the absolute path of the image file that will be created arr (2d np.ndarray) – a 2d numpy array with grey levels

Example

>>> image = openpiv.tools.imread( 'image.bmp' )
>>> image2 = openpiv.tools.negative(image)
>>> imsave( 'negative-image.tif', image2)

openpiv.tools.mark_background(threshold: float, list_img: list, filename: str) → numpy.ndarray[source]¶

marks background

Parameters:	threshold (float) – threshold list_img (list of images) – _description_ filename (str) – image filename to save the mask
Returns:	_description_
Return type:	_type_

openpiv.tools.mark_background2(list_img, filename)[source]¶

openpiv.tools.natural_sort(file_list: List[pathlib.Path]) → List[pathlib.Path][source]¶: Creates naturally sorted list

openpiv.tools.negative(image)[source]¶

Return the negative of an image

image : 2d np.ndarray of grey levels

Returns:	(255-image)
Return type:	2d np.ndarray of grey levels

openpiv.tools.rgb2gray(rgb: numpy.ndarray) → numpy.ndarray[source]¶

converts rgb image to gray

Parameters:	rgb (_type_) – numpy.ndarray, image size, three channels
Returns:	numpy.ndarray of the same shape, one channel
Return type:	gray

openpiv.tools.save(filename: Union[pathlib.Path, str], x: numpy.ndarray, y: numpy.ndarray, u: numpy.ndarray, v: numpy.ndarray, flags: Optional[numpy.ndarray] = None, mask: Optional[numpy.ndarray] = None, fmt: str = '%.4e', delimiter: str = '\t') → None[source]¶

Save flow field to an ascii file.

Parameters:

filename (string) – the path of the file where to save the flow field
x (2d np.ndarray) – a two dimensional array containing the x coordinates of the interrogation window centers, in pixels.
y (2d np.ndarray) – a two dimensional array containing the y coordinates of the interrogation window centers, in pixels.
u (2d np.ndarray) – a two dimensional array containing the u velocity components, in pixels/seconds.
v (2d np.ndarray) – a two dimensional array containing the v velocity components, in pixels/seconds.
flags (2d np.ndarray) – a two dimensional integers array where elements corresponding to vectors: 0 - valid, 1 - invalid (, 2 - interpolated) default: None, will create all valid 0
mask (2d np.ndarray boolean, marks the image masked regions (dynamic and/or static)) – default: None - will be all False

fmt : string: a format string. See documentation of numpy.savetxt for more details.
delimiter : string: character separating columns

Examples

openpiv.tools.save(‘field_001.txt’, x, y, u, v, flags, mask, fmt=’%6.3f’,: delimiter=’ ‘)

openpiv.tools.sorted_unique(array: numpy.ndarray) → numpy.ndarray[source]¶: Creates sorted unique array

openpiv.tools.transform_coordinates(x, y, u, v)[source]¶

Converts coordinate systems from/to the image based / physical based

Input/Output: x,y,u,v

image based is 0,0 top left, x = columns to the right, y = rows downwards and so u,v

physical or right hand one is that leads to the positive vorticity with the 0,0 origin at bottom left to be counterclockwise

openpiv.validation module¶

A module for spurious vector detection.

openpiv.validation.global_std(u: numpy.ndarray, v: numpy.ndarray, std_threshold: int = 5) → numpy.ndarray[source]¶

Eliminate spurious vectors with a global threshold defined by the standard deviation

This validation method tests for the spatial consistency of the data and outliers vector are replaced with NaN (Not a Number) if at least one of the two velocity components is out of a specified global range.

Parameters:	u (2d masked np.ndarray) – a two dimensional array containing the u velocity component. v (2d masked np.ndarray) – a two dimensional array containing the v velocity component. std_threshold (float) – If the length of the vector (actually the sum of squared components) is larger than std_threshold times standard deviation of the flow field, then the vector is treated as an outlier. [default = 3]
Returns:	flag – a boolean array. True elements corresponds to outliers.
Return type:	boolean 2d np.ndarray

openpiv.validation.global_val(u: numpy.ndarray, v: numpy.ndarray, u_thresholds: Tuple[int, int], v_thresholds: Tuple[int, int]) → numpy.ndarray[source]¶

Eliminate spurious vectors with a global threshold.

This validation method tests for the spatial consistency of the data and outliers vector are replaced with Nan (Not a Number) if at least one of the two velocity components is out of a specified global range.

Parameters:	u (2d np.ndarray) – a two dimensional array containing the u velocity component. v (2d np.ndarray) – a two dimensional array containing the v velocity component. u_thresholds (two elements tuple) – u_thresholds = (u_min, u_max). If `u<u_min` or `u>u_max` the vector is treated as an outlier. v_thresholds (two elements tuple) – `v_thresholds = (v_min, v_max)`. If `v<v_min` or `v>v_max` the vector is treated as an outlier.
Returns:	flag – a boolean array. True elements corresponds to outliers.
Return type:	boolean 2d np.ndarray

openpiv.validation.local_median_val(u, v, u_threshold, v_threshold, size=1)[source]¶

Eliminate spurious vectors with a local median threshold.

This validation method tests for the spatial consistency of the data. Vectors are classified as outliers and replaced with Nan (Not a Number) if the absolute difference with the local median is greater than a user specified threshold. The median is computed for both velocity components.

The image masked areas (obstacles, reflections) are marked as masked array:: u = np.ma.masked(u, flag = image_mask)

and it should not be replaced by the local median, but remain masked.

Parameters:	u (2d np.ndarray) – a two dimensional array containing the u velocity component. v (2d np.ndarray) – a two dimensional array containing the v velocity component. u_threshold (float) – the threshold value for component u v_threshold (float) – the threshold value for component v
Returns:	flag – a boolean array. True elements corresponds to outliers.
Return type:	boolean 2d np.ndarray

openpiv.validation.sig2noise_val(s2n: numpy.ndarray, threshold: float = 1.0) → numpy.ndarray[source]¶

Marks spurious vectors if signal to noise ratio is below a specified threshold.

Parameters:	u (2d or 3d np.ndarray) – a two or three dimensional array containing the u velocity component. v (2d or 3d np.ndarray) – a two or three dimensional array containing the v velocity component. s2n (2d np.ndarray) – a two or three dimensional array containing the value of the signal to noise ratio from cross-correlation function. w (2d or 3d np.ndarray) – a two or three dimensional array containing the w (in z-direction) velocity component. threshold (float) – the signal to noise ratio threshold value.
Returns:	flag – a boolean array. True elements corresponds to outliers.
Return type:	boolean 2d np.ndarray

References

1. Keane and R. J. Adrian, Measurement Science & Technology, 1990,
1, 1202-1215.

openpiv.validation.typical_validation(u: numpy.ndarray, v: numpy.ndarray, s2n: numpy.ndarray, settings: PIVSettings) → numpy.ndarray[source]¶

validation using gloabl limits and std and local median,

with a special option of ‘no_std’ for the case of completely uniform shift, e.g. in tests.

see windef.PIVSettings() for the parameters:

MinMaxU : two elements tuple

sets the limits of the u displacment component Used for validation.

MinMaxV : two elements tuple

sets the limits of the v displacment component Used for validation.

std_threshold : float

sets the threshold for the std validation

median_threshold : float

sets the threshold for the median validation

openpiv.widim module¶

openpiv.windef module¶

Created on Fri Oct 4 14:04:04 2019

@author: Theo @modified: Alex, Erich

class openpiv.windef.PIVSettings(filepath_images: Union[pathlib.Path, str] = PosixPath('/home/docs/checkouts/readthedocs.org/user_builds/openpiv/envs/latest/lib/python3.9/site-packages/openpiv/data/test1'), save_path: pathlib.Path = PosixPath('/home/docs/checkouts/readthedocs.org/user_builds/openpiv/envs/latest/lib/python3.9/site-packages/openpiv/data'), save_folder_suffix: str = 'test1', frame_pattern_a: str = 'exp1_001_a.bmp', frame_pattern_b: str = 'exp1_001_b.bmp', roi: Union[Tuple[int, int, int, int], str] = 'full', dynamic_masking_method: Optional[str] = None, dynamic_masking_threshold: float = 0.005, dynamic_masking_filter_size: int = 7, static_mask: Optional[numpy.ndarray] = None, correlation_method: str = 'circular', normalized_correlation: bool = False, windowsizes: Tuple[int, ...] = (64, 32, 16), overlap: Tuple[int, ...] = (32, 16, 8), num_iterations: int = 3, subpixel_method: str = 'gaussian', use_vectorized: bool = False, deformation_method: str = 'symmetric', interpolation_order: int = 3, scaling_factor: float = 1.0, dt: float = 1.0, sig2noise_method: Optional[str] = 'peak2mean', sig2noise_mask: int = 2, sig2noise_threshold: float = 1.0, sig2noise_validate: bool = True, validation_first_pass: bool = True, min_max_u_disp: Tuple = (-30, 30), min_max_v_disp: Tuple = (-30, 30), std_threshold: int = 10, median_threshold: int = 3, median_size: int = 1, replace_vectors: bool = True, smoothn: bool = False, smoothn_p: float = 0.05, filter_method: str = 'localmean', max_filter_iteration: int = 4, filter_kernel_size: int = 2, save_plot: bool = False, show_plot: bool = False, scale_plot: int = 100, show_all_plots: bool = False, invert: bool = False, fmt: str = '%.4e')[source]¶

Bases: object

All the PIV settings for the batch analysis with multi-processing and window deformation. Default settings are set at the initiation

correlation_method = 'circular'¶

deformation_method = 'symmetric'¶

dt = 1.0¶

dynamic_masking_filter_size = 7¶

dynamic_masking_method = None¶: None for no masking ‘edges’ for edges masking, ‘intensity’ for intensity masking

dynamic_masking_threshold = 0.005¶

filepath_images = PosixPath('/home/docs/checkouts/readthedocs.org/user_builds/openpiv/envs/latest/lib/python3.9/site-packages/openpiv/data/test1')¶

filter_kernel_size = 2¶

filter_method = 'localmean'¶

fmt = '%.4e'¶

frame_pattern_a = 'exp1_001_a.bmp'¶

frame_pattern_b = 'exp1_001_b.bmp'¶

interpolation_order = 3¶

invert = False¶

max_filter_iteration = 4¶

median_size = 1¶

median_threshold = 3¶

min_max_u_disp = (-30, 30)¶

min_max_v_disp = (-30, 30)¶

normalized_correlation = False¶

num_iterations = 3¶

overlap = (32, 16, 8)¶

replace_vectors = True¶

roi = 'full'¶

save_folder_suffix = 'test1'¶

save_path = PosixPath('/home/docs/checkouts/readthedocs.org/user_builds/openpiv/envs/latest/lib/python3.9/site-packages/openpiv/data')¶

save_plot = False¶

scale_plot = 100¶

scaling_factor = 1.0¶

show_all_plots = False¶

show_plot = False¶

sig2noise_mask = 2¶

sig2noise_method = 'peak2mean'¶

sig2noise_threshold = 1.0¶

sig2noise_validate = True¶

smoothn = False¶

smoothn_p = 0.05¶

static_mask = None¶

std_threshold = 10¶

subpixel_method = 'gaussian'¶

use_vectorized = False¶

validation_first_pass = True¶

windowsizes = (64, 32, 16)¶

openpiv.windef.create_deformation_field(frame, x, y, u, v, interpolation_order=3)[source]¶

Deform an image by window deformation where a new grid is defined based on the grid and displacements of the previous pass and pixel values are interpolated onto the new grid.

Parameters:

frame (2d np.ndarray, dtype=np.int32) – an two dimensions array of integers containing grey levels of the first frame.
x (2d np.ndarray) – a two dimensional array containing the x coordinates of the interrogation window centers, in pixels.
y (2d np.ndarray) – a two dimensional array containing the y coordinates of the interrogation window centers, in pixels.
u (2d np.ndarray) – a two dimensional array containing the u velocity component, in pixels/seconds.
v (2d np.ndarray) – a two dimensional array containing the v velocity component, in pixels/seconds.
interpolation_order (scalar) – the degree of the interpolation of the B-splines over the rectangular mesh

Returns:

x,y (new grid (after meshgrid))
u,v (deformation field)

openpiv.windef.deform_windows(frame, x, y, u, v, interpolation_order=1, interpolation_order2=3, debugging=False)[source]¶

Deform an image by window deformation where a new grid is defined based on the grid and displacements of the previous pass and pixel values are interpolated onto the new grid.

Parameters:	frame (2d np.ndarray, dtype=np.int32) – an two dimensions array of integers containing grey levels of the first frame. x (2d np.ndarray) – a two dimensional array containing the x coordinates of the interrogation window centers, in pixels. y (2d np.ndarray) – a two dimensional array containing the y coordinates of the interrogation window centers, in pixels. u (2d np.ndarray) – a two dimensional array containing the u velocity component, in pixels/seconds. v (2d np.ndarray) – a two dimensional array containing the v velocity component, in pixels/seconds. interpolation_order (scalar) – the degree of the frame interpolation (deformation) of the image interpolation_order2 (scalar) – the degree of the interpolation of the B-splines over the rectangular mesh
Returns:	a deformed image based on the meshgrid and displacements of the previous pass
Return type:	frame_def

openpiv.windef.first_pass(frame_a, frame_b, settings)[source]¶

First pass of the PIV evaluation.

This function does the PIV evaluation of the first pass. It returns the coordinates of the interrogation window centres, the displacment u and v for each interrogation window as well as the mask which indicates wether the displacement vector was interpolated or not.

Parameters:

frame_a (2d np.ndarray) – the first image
frame_b (2d np.ndarray) – the second image
window_size (int) – the size of the interrogation window
overlap (int) – the overlap of the interrogation window, typically it is window_size/2
subpixel_method (string) – the method used for the subpixel interpolation. one of the following methods to estimate subpixel location of the peak: ‘centroid’ [replaces default if correlation map is negative], ‘gaussian’ [default if correlation map is positive], ‘parabolic’

Returns:

x (2d np.array) – array containg the x coordinates of the interrogation window centres
y (2d np.array) – array containg the y coordinates of the interrogation window centres
u (2d np.array) – array containing the u displacement for every interrogation window
v (2d np.array) – array containing the u displacement for every interrogation window
s2n (2d np.array of the signal to noise ratio)

openpiv.windef.multipass_img_deform(frame_a: numpy.ndarray, frame_b: numpy.ndarray, current_iteration: int, x_old: numpy.ndarray, y_old: numpy.ndarray, u_old: numpy.ndarray, v_old: numpy.ndarray, settings: openpiv.windef.PIVSettings)[source]¶

Multi pass of the PIV evaluation.

This function does the PIV evaluation of the second and other passes. It returns the coordinates of the interrogation window centres, the displacement u, v for each interrogation window as well as the signal to noise ratio array (which is full of NaNs if opted out)

Parameters:

frame_a (2d np.ndarray) – the first image
frame_b (2d np.ndarray) – the second image
window_size (tuple of ints) – the size of the interrogation window
overlap (tuple of ints) – the overlap of the interrogation window, e.g. window_size/2
x_old (2d np.ndarray) – the x coordinates of the vector field of the previous pass
y_old (2d np.ndarray) – the y coordinates of the vector field of the previous pass
u_old (2d np.ndarray) – the u displacement of the vector field of the previous pass in case of the image mask - u_old and v_old are MaskedArrays
v_old (2d np.ndarray) – the v displacement of the vector field of the previous pass
subpixel_method (string) – the method used for the subpixel interpolation. one of the following methods to estimate subpixel location of the peak: ‘centroid’ [replaces default if correlation map is negative], ‘gaussian’ [default if correlation map is positive], ‘parabolic’
interpolation_order (int) – the order of the spline interpolation used for the image deformation
mask_coords (list of x,y coordinates (pixels) of the image mask,) – default is an empty list

Returns:

x (2d np.array) – array containg the x coordinates of the interrogation window centres
y (2d np.array) – array containg the y coordinates of the interrogation window centres
u (2d np.array) – array containing the horizontal displacement for every interrogation window [pixels]
u (2d np.array) – array containing the vertical displacement for every interrogation window it returns values in [pixels]
grid_mask (2d boolean np.array with the image mask in the x,y coordinates)
flags (2D np.array of integers, flags marking 0 - valid, 1 - invalid vectors)

openpiv.windef.piv(settings)[source]¶: the func fuction is the “frame” in which the PIV evaluation is done

openpiv.windef.prepare_images(file_a: pathlib.Path, file_b: pathlib.Path, settings: openpiv.windef.PIVSettings) → Tuple[numpy.ndarray, numpy.ndarray, Optional[numpy.ndarray]][source]¶

prepares two images for the PIV pass

Parameters:	file_a (pathlib.Path) – filename of frame A file_b (pathlib.Path) – filename of frame B settings (_type_) – windef.Settings()

openpiv.windef.simple_multipass(frame_a: numpy.ndarray, frame_b: numpy.ndarray, settings: Optional[PIVSettings] = None) → Tuple[source]¶

_summary_

Parameters:	frame_a (np.ndarray) – frame A image as an array frame_b (np.ndarray) – frame B, settings (Optional["PIVSettings"], optional) – _description_. Defaults to None.
Returns:	_description_
Return type:	Tuple

Module contents¶

openpiv.test()[source]¶